The document discusses mathematical expressions and how to combine them. It defines an expression as a calculation procedure written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined in the same way numbers are, while unlike terms cannot be combined. The simplest expressions are linear expressions of the form ax + b.
The document discusses mathematical expressions and how to combine them. It defines an expression as a calculation procedure written with numbers, variables, and operations. Expressions have terms, with the x-term being the term with a variable and the number term being the constant term. To combine expressions, like terms are combined in the same way numbers are combined. For example, 2x + 3x = 5x and -3x - 5x = -8x. However, unlike terms like x-terms and number terms cannot be combined since they are different types of terms. The overall expression after combining all like terms is called the simplified form.
The document discusses expressions in mathematics. It defines expressions as calculation procedures written with numbers, variables, and operation symbols that calculate outcomes. Expressions can be combined by collecting like terms. Linear expressions take the form of ax + b, where terms can be combined by adding or subtracting the coefficients of the same variable. The example shows combining the terms of the expression 2x - 4 + 9 - 5x.
The document discusses mathematical expressions and how to combine and manipulate them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms can be combined by adding or subtracting their coefficients, while unlike terms cannot be combined. Multiplying an expression distributes the number to each term using the distributive property.
The document discusses mathematical expressions and how to combine them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined by adding or subtracting coefficients in the same way numbers are combined. Unlike terms, such as x-terms and number terms, cannot be combined.
The document discusses variables and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values change depending on the situation. Expressions are made using variables and operations, and evaluation involves replacing variables with input values and finding the output. For example, if x represents the number of apples and each apple costs $2, the expression "2x" gives the total cost for x apples. Evaluation demonstrates calculating the output when specific values are plugged in for the variables.
The document discusses solving linear equations by factoring using an example of determining the number of pizzas ordered. It formulates the problem as the equation 3x + 10 = 34 where x is the number of pizzas. It solves the equation by subtracting 10 from both sides, dividing both sides by 3, and determining that x = 8 pizzas were ordered. The document then provides more details on linear equations, their structure, and their general solution method.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and represent calculation procedures. Evaluation is the process of replacing variables in an expression with input values and calculating the output. This allows expressions to be used to solve real-world problems by modeling relationships between variables.
The document discusses systems of linear equations. It explains that to solve for two unknowns, two equations are needed, and to solve for three unknowns, three equations are required. A system of linear equations consists of two or more linear equations with two or more variables. A solution to the system is a set of numbers for each variable that satisfies all equations. Two examples are provided to demonstrate solving systems of two equations with two unknowns to find the price of items.
The document discusses mathematical expressions and how to combine them. It defines an expression as a calculation procedure written with numbers, variables, and operations. Expressions have terms, with the x-term being the term with a variable and the number term being the constant term. To combine expressions, like terms are combined in the same way numbers are combined. For example, 2x + 3x = 5x and -3x - 5x = -8x. However, unlike terms like x-terms and number terms cannot be combined since they are different types of terms. The overall expression after combining all like terms is called the simplified form.
The document discusses expressions in mathematics. It defines expressions as calculation procedures written with numbers, variables, and operation symbols that calculate outcomes. Expressions can be combined by collecting like terms. Linear expressions take the form of ax + b, where terms can be combined by adding or subtracting the coefficients of the same variable. The example shows combining the terms of the expression 2x - 4 + 9 - 5x.
The document discusses mathematical expressions and how to combine and manipulate them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms can be combined by adding or subtracting their coefficients, while unlike terms cannot be combined. Multiplying an expression distributes the number to each term using the distributive property.
The document discusses mathematical expressions and how to combine them. It defines expressions as calculation procedures written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined by adding or subtracting coefficients in the same way numbers are combined. Unlike terms, such as x-terms and number terms, cannot be combined.
The document discusses variables and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values change depending on the situation. Expressions are made using variables and operations, and evaluation involves replacing variables with input values and finding the output. For example, if x represents the number of apples and each apple costs $2, the expression "2x" gives the total cost for x apples. Evaluation demonstrates calculating the output when specific values are plugged in for the variables.
The document discusses solving linear equations by factoring using an example of determining the number of pizzas ordered. It formulates the problem as the equation 3x + 10 = 34 where x is the number of pizzas. It solves the equation by subtracting 10 from both sides, dividing both sides by 3, and determining that x = 8 pizzas were ordered. The document then provides more details on linear equations, their structure, and their general solution method.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and represent calculation procedures. Evaluation is the process of replacing variables in an expression with input values and calculating the output. This allows expressions to be used to solve real-world problems by modeling relationships between variables.
The document discusses systems of linear equations. It explains that to solve for two unknowns, two equations are needed, and to solve for three unknowns, three equations are required. A system of linear equations consists of two or more linear equations with two or more variables. A solution to the system is a set of numbers for each variable that satisfies all equations. Two examples are provided to demonstrate solving systems of two equations with two unknowns to find the price of items.
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Specific inequalities are shown graphically on the number line. The properties of inequalities, such as adding or multiplying the same quantity to both sides preserving the inequality, are also covered.
6.1 system of linear equations and matricesmath260
The document discusses solving systems of linear equations using the elimination method. It provides an example of a system with two unknowns, x and y, representing the cost of a hamburger and fries. The system is translated into two equations, which are then subtracted to eliminate one variable. The resulting single equation is solved for the remaining variable, which is substituted back into the original system to solve for both unknowns. A second example system with three unknowns, x, y, and z, is also given and the elimination method is shown being applied step-by-step to reduce the system.
The document discusses systems of linear equations. It defines a system of linear equations as a collection of two or more linear equations with two or more variables. A solution to a system is a set of numbers, one for each variable, that satisfies all equations simultaneously. The document provides an example system involving the costs of hamburgers and fries to illustrate how to solve a system of two equations with two unknowns.
The document provides an example of using a linear equation to solve a word problem about ordering pizzas. It shows setting up the equation 3x + 10 = 34 to represent the total cost of x pizzas with a $10 delivery fee when the total is $34. It then works through the step-by-step process of solving for x, finding that x = 8 pizzas. The document goes on to define linear equations as those containing only first degree terms, and explains that linear equations are easy to solve by manipulating the equation to find the value of the variable.
The document discusses linear equations and how to solve them. It begins by providing an example of solving a multi-step linear equation to find the number of pizzas ordered given the total cost. It then defines linear equations as those containing only first degree terms of the variable and no higher powers. The document states that linear equations are easy to solve by manipulating the equation to isolate the variable. It provides examples of single-step linear equations and explains the basic principle is to apply the opposite operation to both sides to isolate the variable.
The document discusses fractions and their properties. It defines fractions as numbers of the form p/q where p and q are natural numbers. Fractions represent parts of a whole, for example 3/6 represents 3 out of 6 equal slices of a pizza. The numerator is the number on top and represents the parts, while the denominator on bottom represents the total parts of the whole. Equivalent fractions like 1/2, 2/4, and 3/6 represent the same quantity. Dividing the numerator and denominator by a common factor results in an equivalent fraction. The denominator of a fraction cannot be zero, as this results in an undefined fraction.
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
The document discusses systems of linear equations. It begins by explaining that to solve for one unknown quantity, one piece of information is needed, and to solve for two unknowns, two pieces of information are needed. This leads to systems of linear equations, which are collections of two or more linear equations with two or more variables. A solution to a system is a set of numbers for the variables that satisfies all equations. An example system is provided. The document then works through an example problem about the cost of hamburgers and salads to demonstrate solving a system of linear equations.
49 factoring trinomials the ac method and making listsalg1testreview
The document discusses factoring trinomials. It explains that trinomials are polynomials of the form ax2 + bx + c. There are two types of trinomials: those that are factorable, which can be written as the product of two binomials, and those that are not factorable. The ac-method uses tables to determine whether a trinomial is factorable or not by finding values for u and v that fit the table. If values are found, the trinomial can be factored using the grouping method. An example factors the trinomial x2 - x - 6 by grouping.
13 fractions, multiplication and divisin of fractionsalg1testreview
The document defines fractions as numbers of the form p/q where p and q are whole numbers not equal to 0. Fractions represent parts of a whole, for example the fraction 3/6 represents 3 out of 6 equal slices of a pizza. The top number is called the numerator and represents the number of parts, while the bottom number is the denominator and represents the total number of equal parts the whole was divided into. Examples are provided to demonstrate calculating fractional amounts of groups of items. It is noted that fractions with a denominator of 0 are undefined in mathematics.
The document discusses operations that can be performed on polynomial expressions. It defines terms and like-terms in polynomials, and explains that like-terms can be combined while unlike terms cannot. It provides examples of combining like-terms, expanding polynomials using the distributive property, multiplying terms and polynomials, and simplifying the results.
The document discusses addition and subtraction. It defines addition as combining two quantities A and B to obtain a sum S, where A and B are the addends. To add two numbers, one lines them up vertically according to place value and adds the digits from right to left, carrying when necessary. Subtraction is defined as taking away or undoing addition. To subtract, one lines numbers up vertically and subtracts the digits from right to left, borrowing when needed. Real-life word problems involving addition and subtraction are translated into mathematical expressions by clarifying which quantity is being added to or subtracted from the other.
The document discusses natural numbers and factors. It defines natural numbers as whole numbers used to count items. A factor of a natural number x is another natural number y that x can be divided by evenly. The document then provides an example of listing the factors and multiples of 12. It also discusses prime numbers as numbers with only two factors, 1 and the number itself. The document concludes with definitions of factoring a number and exponents.
The document discusses solving word problems by translating English phrases into mathematical operations. It provides key words that represent different operations such as "+" for add and "-" for subtract. It then gives examples of word problems and shows how to represent the problems mathematically by defining variables, identifying unknown values, and relating other values to the variables. The goal is to break down word problems into clear mathematical expressions.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
32 multiplication and division of decimalsalg1testreview
The document discusses how to multiply multi-digit numbers by treating it as multiple single-digit multiplication problems. It shows working through an example problem step-by-step, multiplying 47 x 685. Each digit is multiplied by the bottom number and the results are placed in columns with carrying as needed. The columns are then added to obtain the final answer. The process is similar for multiplying decimal numbers, ignoring the decimal points during the multiplication and then placing the decimal point in the correct position in the final product.
The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents:
1) The multiply-add rule: ANAK = AN+K
2) The divide-subtract rule: AN/AK = AN-K
3) The power-multiply rule: (AN)K = ANK
4) Additional rules including that A0 = 1 and A-K = 1/AK
The document discusses exponents and the rules for working with them. Exponents describe the number of times a base is multiplied by itself. The main rules covered are:
- The multiply-add rule: When the same base has two exponents, you add the exponents and multiply the results, written as ANAK = AN+K
- The divide-subtract rule: When dividing exponents with the same base, you subtract the exponents and divide the results, written as AN/AK = AN-K
Examples are provided to demonstrate calculating exponents using these rules.
Light can travel 186,000 miles in one minute. In scientific notation, this is 1.86x105 miles.
The document discusses multiplying polynomials and binomials. It introduces the box method and the F.O.I.L (First, Outer, Inner, Last) method for multiplying binomials. F.O.I.L involves distributing the first term of one binomial with each term of the other binomial. The sign of the middle and last terms depends on whether the signs of the last terms of each binomial are the same or different.
The document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the original fractional ratio using whole numbers. An example demonstrates taking a ratio of 3/4 cups sugar to 2/3 cups flour and rewriting it as 9:8 cups sugar to flour using cross multiplication. The document also notes cross multiplication can be used to compare two fractions, with the larger product corresponding to the larger fraction.
The document defines percentages as expressing "how many out of 100" and can be written as a fraction with the percentage symbol or as a decimal. It provides examples of common percentages like 1% = 1/100, 5% = 1/20, 10% = 1/10, 25% = 1/4, and 50% = 1/2. The document also works through examples of calculating percentages of a total amount, like finding 3/4 of $100 is $75, or that 45% of 60 pieces of candy is 27 pieces. Finally, it lists some important percentages that relate to coins: 5% = 1/20 for nickels, 10% = 1/10 for dimes, and 25%
The document discusses polynomial expressions in mathematics. It defines a polynomial as the sum of monomial terms, where a monomial is a number multiplied by a variable raised to a non-negative integer power. The degree of a polynomial is defined as the highest exponent among its monomial terms. Several examples of evaluating monomials and polynomials are provided by substituting specific values for variables.
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the position of two numbers on the real number line, with the number farther to the right said to be greater than the number on the left. Specific inequalities are shown graphically on the number line. The properties of inequalities, such as adding or multiplying the same quantity to both sides preserving the inequality, are also covered.
6.1 system of linear equations and matricesmath260
The document discusses solving systems of linear equations using the elimination method. It provides an example of a system with two unknowns, x and y, representing the cost of a hamburger and fries. The system is translated into two equations, which are then subtracted to eliminate one variable. The resulting single equation is solved for the remaining variable, which is substituted back into the original system to solve for both unknowns. A second example system with three unknowns, x, y, and z, is also given and the elimination method is shown being applied step-by-step to reduce the system.
The document discusses systems of linear equations. It defines a system of linear equations as a collection of two or more linear equations with two or more variables. A solution to a system is a set of numbers, one for each variable, that satisfies all equations simultaneously. The document provides an example system involving the costs of hamburgers and fries to illustrate how to solve a system of two equations with two unknowns.
The document provides an example of using a linear equation to solve a word problem about ordering pizzas. It shows setting up the equation 3x + 10 = 34 to represent the total cost of x pizzas with a $10 delivery fee when the total is $34. It then works through the step-by-step process of solving for x, finding that x = 8 pizzas. The document goes on to define linear equations as those containing only first degree terms, and explains that linear equations are easy to solve by manipulating the equation to find the value of the variable.
The document discusses linear equations and how to solve them. It begins by providing an example of solving a multi-step linear equation to find the number of pizzas ordered given the total cost. It then defines linear equations as those containing only first degree terms of the variable and no higher powers. The document states that linear equations are easy to solve by manipulating the equation to isolate the variable. It provides examples of single-step linear equations and explains the basic principle is to apply the opposite operation to both sides to isolate the variable.
The document discusses fractions and their properties. It defines fractions as numbers of the form p/q where p and q are natural numbers. Fractions represent parts of a whole, for example 3/6 represents 3 out of 6 equal slices of a pizza. The numerator is the number on top and represents the parts, while the denominator on bottom represents the total parts of the whole. Equivalent fractions like 1/2, 2/4, and 3/6 represent the same quantity. Dividing the numerator and denominator by a common factor results in an equivalent fraction. The denominator of a fraction cannot be zero, as this results in an undefined fraction.
The document discusses how to solve radical equations by squaring both sides of the equation repeatedly to remove radicals. Key steps include:
1) Isolating the radical term to one side of the equation before squaring.
2) Using the identity (a ± b)2 = a2 ± 2ab + b2 to expand squared terms.
3) Squaring both sides and solving the resulting non-radical equation for the variable.
4) Checking that solutions satisfy the original radical equation. Examples demonstrate these techniques.
The document discusses systems of linear equations. It begins by explaining that to solve for one unknown quantity, one piece of information is needed, and to solve for two unknowns, two pieces of information are needed. This leads to systems of linear equations, which are collections of two or more linear equations with two or more variables. A solution to a system is a set of numbers for the variables that satisfies all equations. An example system is provided. The document then works through an example problem about the cost of hamburgers and salads to demonstrate solving a system of linear equations.
49 factoring trinomials the ac method and making listsalg1testreview
The document discusses factoring trinomials. It explains that trinomials are polynomials of the form ax2 + bx + c. There are two types of trinomials: those that are factorable, which can be written as the product of two binomials, and those that are not factorable. The ac-method uses tables to determine whether a trinomial is factorable or not by finding values for u and v that fit the table. If values are found, the trinomial can be factored using the grouping method. An example factors the trinomial x2 - x - 6 by grouping.
13 fractions, multiplication and divisin of fractionsalg1testreview
The document defines fractions as numbers of the form p/q where p and q are whole numbers not equal to 0. Fractions represent parts of a whole, for example the fraction 3/6 represents 3 out of 6 equal slices of a pizza. The top number is called the numerator and represents the number of parts, while the bottom number is the denominator and represents the total number of equal parts the whole was divided into. Examples are provided to demonstrate calculating fractional amounts of groups of items. It is noted that fractions with a denominator of 0 are undefined in mathematics.
The document discusses operations that can be performed on polynomial expressions. It defines terms and like-terms in polynomials, and explains that like-terms can be combined while unlike terms cannot. It provides examples of combining like-terms, expanding polynomials using the distributive property, multiplying terms and polynomials, and simplifying the results.
The document discusses addition and subtraction. It defines addition as combining two quantities A and B to obtain a sum S, where A and B are the addends. To add two numbers, one lines them up vertically according to place value and adds the digits from right to left, carrying when necessary. Subtraction is defined as taking away or undoing addition. To subtract, one lines numbers up vertically and subtracts the digits from right to left, borrowing when needed. Real-life word problems involving addition and subtraction are translated into mathematical expressions by clarifying which quantity is being added to or subtracted from the other.
The document discusses natural numbers and factors. It defines natural numbers as whole numbers used to count items. A factor of a natural number x is another natural number y that x can be divided by evenly. The document then provides an example of listing the factors and multiples of 12. It also discusses prime numbers as numbers with only two factors, 1 and the number itself. The document concludes with definitions of factoring a number and exponents.
The document discusses solving word problems by translating English phrases into mathematical operations. It provides key words that represent different operations such as "+" for add and "-" for subtract. It then gives examples of word problems and shows how to represent the problems mathematically by defining variables, identifying unknown values, and relating other values to the variables. The goal is to break down word problems into clear mathematical expressions.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
32 multiplication and division of decimalsalg1testreview
The document discusses how to multiply multi-digit numbers by treating it as multiple single-digit multiplication problems. It shows working through an example problem step-by-step, multiplying 47 x 685. Each digit is multiplied by the bottom number and the results are placed in columns with carrying as needed. The columns are then added to obtain the final answer. The process is similar for multiplying decimal numbers, ignoring the decimal points during the multiplication and then placing the decimal point in the correct position in the final product.
The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents:
1) The multiply-add rule: ANAK = AN+K
2) The divide-subtract rule: AN/AK = AN-K
3) The power-multiply rule: (AN)K = ANK
4) Additional rules including that A0 = 1 and A-K = 1/AK
The document discusses exponents and the rules for working with them. Exponents describe the number of times a base is multiplied by itself. The main rules covered are:
- The multiply-add rule: When the same base has two exponents, you add the exponents and multiply the results, written as ANAK = AN+K
- The divide-subtract rule: When dividing exponents with the same base, you subtract the exponents and divide the results, written as AN/AK = AN-K
Examples are provided to demonstrate calculating exponents using these rules.
Light can travel 186,000 miles in one minute. In scientific notation, this is 1.86x105 miles.
The document discusses multiplying polynomials and binomials. It introduces the box method and the F.O.I.L (First, Outer, Inner, Last) method for multiplying binomials. F.O.I.L involves distributing the first term of one binomial with each term of the other binomial. The sign of the middle and last terms depends on whether the signs of the last terms of each binomial are the same or different.
The document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the original fractional ratio using whole numbers. An example demonstrates taking a ratio of 3/4 cups sugar to 2/3 cups flour and rewriting it as 9:8 cups sugar to flour using cross multiplication. The document also notes cross multiplication can be used to compare two fractions, with the larger product corresponding to the larger fraction.
The document defines percentages as expressing "how many out of 100" and can be written as a fraction with the percentage symbol or as a decimal. It provides examples of common percentages like 1% = 1/100, 5% = 1/20, 10% = 1/10, 25% = 1/4, and 50% = 1/2. The document also works through examples of calculating percentages of a total amount, like finding 3/4 of $100 is $75, or that 45% of 60 pieces of candy is 27 pieces. Finally, it lists some important percentages that relate to coins: 5% = 1/20 for nickels, 10% = 1/10 for dimes, and 25%
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of evaluating expressions involving multiplication, division, addition, subtraction, grouping symbols and exponents. The key steps are to perform operations within grouping symbols from the innermost out, then multiplication and division from left to right, followed by addition and subtraction from left to right. Setting clear rules for order of operations ensures the correct solution is obtained.
31 decimals, addition and subtraction of decimalsalg-ready-review
This document introduces decimals by using an analogy of a cash register holding coins of various values. It explains that decimals allow tracking of smaller quantities by including base-10 fractions in the number system. It assumes the US Treasury makes fictional smaller value coins like "itties" and "bitties", then demonstrates writing decimal numbers as representations of coins in different slots of a cash register. Finally, it provides steps for comparing decimal values by lining up numbers at the decimal point and determining the largest by the digits from left to right.
Flor Stella Fraile Cano lives in Ubate and has a file card number of 748919. In January, she went to La Vega with her boyfriend where the weather was warm and sunny. They visited a country house and pool, where Flor swam but did not go into the city or dance at a club. The best part was traveling by bus.
34 conversion between decimals, fractions and percentagesalg-ready-review
The document discusses the conversion between decimals, fractions, and percentages. It states that fractions, decimals, and percentages are different ways to express quantities. Fractions provide instructions to divide a whole into parts, while decimals standardize fractions to powers of 10 to make addition and subtraction easier. Percentages express a quantity as a ratio out of 100. The document then provides examples and steps for converting between these representations.
The document discusses how to multiply multi-digit decimal numbers. It explains that multiplication of multi-digit numbers is done by multiplying the digits in place value, starting from the ones place. The results are recorded and carried over as needed. It provides a step-by-step example of 47 x 6, showing how each digit is multiplied and the results carried to the next place value. It notes that the same process is followed for decimals, but the decimal point is placed in the final product so that the total number of decimal places is correct.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and evaluation involves replacing the variables in an expression with input values and calculating the output. The input values replace the variables within parentheses, and the process of evaluation finds the output. Several examples are provided to demonstrate evaluating different expressions by replacing variables with given input values.
The document provides examples and explanations for solving linear equations with one step. It defines a linear equation as one where both sides are linear expressions, such as 3x + 10 = 34, and not containing higher powers of x. To solve a one-step linear equation, the goal is to isolate the variable x on one side by applying the opposite operation to both sides, such as adding 3 to both sides of x - 3 = 12 to get x = 15. Worked examples are provided for solving equations of the form x ± a = b and cx = d.
The document provides examples of solving linear equations. It explains that a linear equation is one where the expressions on both sides are linear, such as 3x + 10 = 34. An example problem involves calculating the cost of x pizzas including delivery. By setting up and solving the equation 3x + 10 = 34, it is determined that x = 8 pizzas. The document discusses manipulating linear equations through steps like subtraction to solve for the variable.
The document discusses solving linear equations using examples of ordering pizzas. It explains that a linear equation contains linear expressions on both sides, such as 3x + 10 = 34, and can be solved by manipulating the equation through steps like subtraction to find the value of x that makes both sides equal. For example, in the equation 3x + 10 = 34, subtracting 10 from both sides and dividing both sides by 3 reveals that x = 8 is the solution.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions involving variables and operations. Polynomial expressions are algebraic expressions that can be written in the form anxn + an-1xn-1 + ... + a1x + a0, where the ai coefficients are numbers. The document gives examples of factoring polynomials using formulas like a3b3 = (ab)(a2ab + b2). Factoring polynomials makes it easier to calculate outputs and simplify expressions for operations like addition and subtraction.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions and operations with polynomials, such as factoring polynomials. Factoring polynomials makes it easier to calculate outputs, simplify rational expressions, and solve equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). It notes that factoring polynomials is useful for evaluating polynomial expressions more easily, as demonstrated by an example evaluating the factored form of 2x3 - 5x2 + 2x for various values of x.
The document discusses mathematical expressions and polynomials. It provides examples of algebraic expressions and operations with polynomials, such as factoring polynomials. Factoring polynomials makes it easier to calculate outputs, simplify rational expressions, and solve equations. One example factors the polynomial 2x3 - 5x2 + 2x and shows it is easier to evaluate the factored form for different values of x than the original polynomial. The key purposes of factoring polynomials are to simplify calculations and operations.
The document discusses mathematical expressions and algebraic expressions. It provides examples of algebraic expressions like 3x^2 - 2x + 4 and explains how to perform operations on polynomial expressions, like factoring 64x^3 + 125 as (4x + 5)(16x^2 - 20x + 25). The key purposes of factoring polynomials are stated as making it easier to calculate outputs, simplify rational expressions, and solve equations. An example is given to evaluate the factored expression 2x^3 - 5x^2 + 2x for various values of x.
The document discusses expressions and polynomials. It provides examples of algebraic expressions and operations that can be performed on polynomials, such as factoring. Factoring polynomials is useful for easier evaluation, simplifying rational expressions, and solving equations. One example factors the polynomial 64x3 + 125 into (4x + 5)(16x2 - 20x + 25). Factoring the polynomial 2x3 - 5x2 + 2x is recommended before evaluating it for specific values of x.
The document discusses expressions and equations. It provides an example of calculating the total cost of ordering x pizzas from Pizza Grande using the expression "8x + 10". It then shows how to solve the equation "8x + 10 = 810" to determine that x = 100 pizzas were ordered. The document explains that equations set two expressions equal and solving an equation means finding the value of the variable that makes the equation true. It distinguishes between linear and quadratic equations.
The document explains multiplication and how it is represented by the multiplication sign. It defines multiplication as simplifying the notation for repeatedly adding the same quantity. For example, 3 x 2 represents 3 copies of 2, or 2 + 2 + 2, which equals 6. The document notes that multiplication is commutative, so the order of the factors does not matter. It introduces the terms factors, product, and the vertical format for multiplying larger numbers. It provides examples of multiplying multi-digit numbers and carrying values between steps. Key features of the multiplication table are also summarized.
This document discusses linear equations in one variable. It defines linear equations as those involving single variables with the highest power being 1. It presents rules for solving linear equations, including adding, subtracting, multiplying, or dividing the same quantity to both sides. Transposition as a method is explained, where terms change signs when shifted between sides of an equation. Examples of solving linear equations are provided. The document also discusses applying linear equations to word problems by setting up the equation based on the problem and solving for the unknown variable. Several examples of solving word problems involving linear equations are presented.
The document discusses various properties of real numbers including the commutative, associative, identity, inverse, zero, and distributive properties. It also covers topics such as combining like terms, translating word phrases to algebraic expressions, and simplifying algebraic expressions. Examples are provided to illustrate each concept along with explanations of key terms like coefficients, variables, and like terms.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and operations, and evaluation involves replacing variables in an expression with input values to obtain the output. For example, if x represents the number of apples and the cost is $2 each, the expression "2x" gives the total cost for x apples.
1. An algebraic expression is a combination of numbers, variables, and operation symbols. It can be classified as a monomial, binomial, or trinomial based on the number of terms.
2. Like terms contain the same variables raised to the same powers, while unlike terms do not. Multiplication of algebraic expressions follows rules such as the product of like signs being positive and unlike signs being negative.
3. There are special product identities for multiplying binomials and factoring algebraic expressions through grouping and finding greatest common factors. Division of algebraic expressions also follows rules regarding the signs of the quotient.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and represent calculation procedures. Evaluation is the process of replacing variables in an expression with input values and calculating the output. Examples are provided to demonstrate how to evaluate expressions by substituting values for variables enclosed in parentheses.
factoring trinomials the ac method and making listsmath260
This document discusses factoring trinomials. It defines a trinomial as a three-term polynomial of the form ax2 + bx + c, where a, b, and c are numbers. There are two types of trinomials: those that are factorable, which can be written as the product of two binomials, and those that are prime or unfactorable. The document outlines the basic rules for factoring trinomials and provides an example of expanding a binomial expression into a trinomial.
The document discusses systems of linear equations. It provides examples to illustrate that we need as many equations as unknowns to solve for the unknown variables. For a system with two unknowns, we need two equations; for three unknowns, we need three equations. The document also gives examples of setting up and solving systems of linear equations to find unknown costs given information about total costs.
This chapter discusses using algebra to represent and solve problems involving addition, subtraction, and finding patterns and rules. It includes the following key points:
- Lesson 3-1 covers writing and evaluating expressions with variables and addition/subtraction.
- Lesson 3-2 explains how to solve addition and subtraction equations mentally without using models.
- Lesson 3-3 introduces identifying extra and missing information in word problems in order to write and solve the correct equations.
- Lesson 3-4 teaches finding patterns in tables and writing rules as equations that can be used to determine future terms in the pattern.
Linear Equations and Inequalities in One Variablemisey_margarette
The document discusses linear equations and inequalities in one variable. It defines linear equations and inequalities, and describes methods for solving them including: guess-and-check, cover-up, and working backwards. It also covers properties of equality and inequality, and provides examples of solving linear equations and inequalities using these properties and graphical representations of solution sets on number lines.
This document provides examples of solving various types of linear equations and inequalities in one variable. It demonstrates solving equations and inequalities using properties of equality and inequality, such as adding or subtracting the same quantity to both sides. It also discusses representing and solving
This document contains a sample algebra test with 24 multiple choice questions and solutions. Each question is followed by an answer option and a link to the solution. The solutions provide step-by-step workings to arrive at the correct answer. The test covers topics such as fractions, operations with fractions, percentages, word problems involving percentages, and solving simple equations. An interactive PowerPoint file is also included but viewing the slides is not interactive when using Slideshare.
The rectangular coordinate system represents points in a plane using perpendicular axes (x-axis and y-axis) that intersect at the origin (0,0). Each point is assigned an ordered pair (x,y) where x is the distance from the origin on the x-axis and y is the distance from the origin on the y-axis. The system divides the plane into four quadrants based on whether the x and y values are positive or negative. The rectangular coordinate system allows any point in the plane to be uniquely addressed using its x and y coordinates.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to the right of 0 and negative numbers to the left. Any number to the right of another number on the number line is defined as greater than the number to its left. Intervals on the number line, denoted by a < x < b, represent all the numbers between a and b. Examples are provided to illustrate drawing intervals on the number line.
The document discusses right triangles and the Pythagorean theorem. It defines a right triangle as one with a 90-degree angle, and labels the sides as the hypotenuse (the side opposite the right angle) and the two legs. It presents the Pythagorean theorem, which states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The document provides an example of using the theorem to calculate the height of a wall given the length of a leaning ladder. It also defines the square root and how it relates to finding the length of a side of a triangle based on the Pythagorean theorem.
The document discusses basic geometric shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed by connecting line segments end to end. Triangles have three sides and their perimeter is calculated as the sum of the three side lengths. Specific types of triangles like equilateral triangles are discussed. Rectangles are four-sided polygons with right angles, and squares are rectangles with four equal sides. Formulas are provided for calculating the perimeters of squares and rectangles based on their side lengths. Some example problems demonstrate applying these concepts and formulas to calculate perimeters of fenced or roped areas composed of multiple shapes.
The document discusses ratios, proportions, and how to solve proportional equations. It defines a ratio as two related quantities stated side by side, and gives the example of a 3:4 ratio of eggs to flour in a recipe. Proportions are defined as equal ratios. The key steps to solve proportional equations are: 1) write the ratios as fractions set equal to each other, 2) use cross-multiplication to convert the proportions into regular equations, and 3) solve the resulting equation using algebraic techniques. An example problem demonstrates these steps to solve a proportional equation for the variable x.
The document discusses exponents and rules for working with them. It defines exponents as the number of times a base is used as a factor in a repetitive multiplication. The main rules covered are:
- The multiply-add rule, which states that ANAK = AN+K
- The divide-subtract rule, which states that AN/AK = AN-K
Examples are provided to demonstrate calculating exponents and applying the rules.
23 multiplication and division of signed numbersalg-ready-review
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use the following rules for the sign of the product: two numbers with the same sign yield a positive product; two numbers with opposite signs yield a negative product. It also discusses how multiplication is implied in algebra without an explicit operation symbol between terms.
The document discusses the Pythagorean theorem and square roots. It defines a right triangle as having one 90 degree angle. The Pythagorean theorem states that for a right triangle with sides a, b, c, where c is the hypotenuse opposite the right angle, a^2 + b^2 = c^2. An example uses the theorem to calculate the height of a wall given the length of a ladder leaning against it. Square roots are then introduced, with the square root of a number x defined as the positive number that produces x when squared.
The document discusses finding the least common multiple (LCM) of numbers. It defines the LCM as the smallest number that is a multiple of all the given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The preferred method when the LCM is large is to construct it by fully factorizing each number into prime factors and taking the highest power of each prime factor.
The document discusses rules for multiplying fractions. It states that to multiply fractions, one should multiply the numerators and multiply the denominators, canceling terms when possible. It then provides examples, such as multiplying 12/25 * 15/8, simplifying to 9/10. It also notes that word problems involving fractions of a quantity can often be solved by translating them into fraction multiplications.
The document defines fractions as numbers of the form p/q where p and q are whole numbers not equal to 0. Fractions represent parts of a whole, for example 3/6 of a pizza. The top number is the numerator and represents the number of parts, while the bottom number is the denominator and represents the total number of equal parts the whole was divided into. Calculations with fractions involve dividing the whole into the number of parts in the denominator and taking the number of parts indicated by the numerator. Whole numbers can be viewed as having a denominator of 1. Dividing by 0 is undefined in mathematics.
The document defines and explains prime factorization. It states that natural numbers can be divided into multiples and factors. A prime number is only divisible by 1 and itself. To find the prime factorization of a number, it is broken down into prime number factors. For example, the prime factorization of 12 is 2 * 2 * 3, as these are all prime numbers that multiply to 12. Finding the complete prime factorization involves writing the number as a product of only prime numbers.
The document discusses addition and subtraction. It defines addition as combining two quantities A and B to obtain a sum S, where A and B are the addends. To add two numbers, one lines them up vertically according to place value and adds the digits from right to left, carrying when necessary. Subtraction is defined as taking away or undoing an addition. To subtract, one lines numbers up vertically and subtracts the digits from right to left, borrowing when needed. Examples of adding and subtracting multi-digit numbers are provided.
This document provides an overview of the topics covered in an Algebra-Ready placement test, including numbers and arithmetic operations with fractions, decimals, percentages, and signed numbers; algebra basics like expressions, equations, exponents and variables; and geometry concepts involving basic shapes, the Pythagorean theorem, and coordinate systems. Fractions covered include arithmetic operations, LCM, LCD, addition/subtraction, and cross-multiplication. Signed numbers include addition, subtraction, multiplication, division and order of operations. Decimals cover addition, subtraction, multiplication, division and conversions between decimals and fractions.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
2. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
Expressions
3. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
Expressions
4. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
Expressions
5. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
Expressions
6. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $..
Expressions
7. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $..
Expressions
Formulas such as “3x” or “3x + 10” are called expressions
in mathematics.
8. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $..
Expressions
Formulas such as “3x” or “3x + 10” are called expressions
in mathematics. Mathematical expressions are calculation
procedures and they are written with numbers, variables,
and operation symbols.
9. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $..
Expressions
Formulas such as “3x” or “3x + 10” are called expressions
in mathematics. Mathematical expressions are calculation
procedures and they are written with numbers, variables,
and operation symbols. Expressions calculate outcomes.
10. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $..
Expressions
Formulas such as “3x” or “3x + 10” are called expressions
in mathematics. Mathematical expressions are calculation
procedures and they are written with numbers, variables,
and operation symbols. Expressions calculate outcomes.
The simplest type of expressions are of the form ax + b where
a and b are numbers.
11. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $..
Expressions
Formulas such as “3x” or “3x + 10” are called expressions
in mathematics. Mathematical expressions are calculation
procedures and they are written with numbers, variables,
and operation symbols. Expressions calculate outcomes.
The simplest type of expressions are of the form ax + b where
a and b are numbers. These are called linear expressions.
12. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $.
Expressions
Formulas such as “3x” or “3x + 10” are called expressions
in mathematics. Mathematical expressions are calculation
procedures and they are written with numbers, variables,
and operation symbols. Expressions calculate outcomes.
The simplest type of expressions are of the form ax + b where
a and b are numbers. These are called linear expressions.
The expressions “3x” or “3x + 10” are linear,
13. Example A.
a. We order pizzas from Pizza Grande. Each pizza is $3.
How much would it cost for 4 pizzas? For x pizzas?
For 4 pizzas, it would cost 3 * 4 = $12,
for x pizzas it would cost 3 * x = $3x.
b. There is $10 delivery charge. How much would it cost us
in total if we want the x pizzas delivered?
In total, it would be 3x + 10 in $.
Expressions
Formulas such as “3x” or “3x + 10” are called expressions
in mathematics. Mathematical expressions are calculation
procedures and they are written with numbers, variables,
and operation symbols. Expressions calculate outcomes.
The simplest type of expressions are of the form ax + b where
a and b are numbers. These are called linear expressions.
The expressions “3x” or “3x + 10” are linear,
the expressions “x2 + 1” or “1/x” are not linear.
19. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms.
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
20. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms. Hence 2x + 3x = 5x.
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
21. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms. Hence 2x + 3x = 5x.
We combine positive x-terms and negative x-terms the same
way we combine signed numbers.
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
22. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms. Hence 2x + 3x = 5x.
We combine positive x-terms and negative x-terms the same
way we combine signed numbers. Hence –3x – 5x = –8x.
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
23. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms. Hence 2x + 3x = 5x.
We combine positive x-terms and negative x-terms the same
way we combine signed numbers. Hence –3x – 5x = –8x.
The number-terms may be combined such as 3 – 8 = –5.
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
24. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms. Hence 2x + 3x = 5x.
We combine positive x-terms and negative x-terms the same
way we combine signed numbers. Hence –3x – 5x = –8x.
The number-terms may be combined such as 3 – 8 = –5.
However just as 2 apple + 3 banana = 2 apple + 3 banana,
(i.e. they can’t be combined),
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
25. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms. Hence 2x + 3x = 5x.
We combine positive x-terms and negative x-terms the same
way we combine signed numbers. Hence –3x – 5x = –8x.
The number-terms may be combined such as 3 – 8 = –5.
However just as 2 apple + 3 banana = 2 apple + 3 banana,
(i.e. they can’t be combined), x-terms can't be combined with
number terms because they are different type of items.
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
26. Just as 2 apples + 3 apples = 5 apples we may combine two
x-terms. Hence 2x + 3x = 5x.
We combine positive x-terms and negative x-terms the same
way we combine signed numbers. Hence –3x – 5x = –8x.
The number-terms may be combined such as 3 – 8 = –5.
However just as 2 apple + 3 banana = 2 apple + 3 banana,
(i.e. they can’t be combined), x-terms can't be combined with
number terms because they are different type of items.
Hence 2 + 3x stays as 2 + 3x (not 5x).
Expressions
Combining Expressions
Given an expression, each addend is called a term. There are
two terms in ax + b.
the x-term the number term or the constant term
30. For the x-term ax, the number “a” is called the coefficient of
the term.
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
31. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient.
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
32. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
33. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
34. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
We may multiply a number with an expression and expand the
result by the distributive law.
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
35. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
We may multiply a number with an expression and expand the
result by the distributive law.
Distributive Law
A(B ± C) = AB ± AC = (B ± C)A
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
36. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
We may multiply a number with an expression and expand the
result by the distributive law.
Distributive Law
A(B ± C) = AB ± AC = (B ± C)A
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
Example C. Expand then simplify.
a. –5(2x – 4)
37. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
We may multiply a number with an expression and expand the
result by the distributive law.
Distributive Law
A(B ± C) = AB ± AC = (B ± C)A
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
Example C. Expand then simplify.
a. –5(2x – 4)
= –5(2x) – (–5)(4)
38. For the x-term ax, the number “a” is called the coefficient of
the term. When we multiply a number with an x-term, we
multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,
and –2(–4x) = (–2)(–4)x = 8x.
We may multiply a number with an expression and expand the
result by the distributive law.
Distributive Law
A(B ± C) = AB ± AC = (B ± C)A
Expressions
Example B. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
Example C. Expand then simplify.
a. –5(2x – 4)
= –5(2x) – (–5)(4)
= –10x + 20
46. b. 3(2x – 4) + 2(4 – 5x) expand
= 6x – 12 + 8 – 10x
= –4x – 4
Expressions
Distributive law gives us the option of expanding the content of
the parentheses i.e. extract items out of containers.
47. b. 3(2x – 4) + 2(4 – 5x) expand
= 6x – 12 + 8 – 10x
= –4x – 4
Expressions
Distributive law gives us the option of expanding the content of
the parentheses i.e. extract items out of containers.
Example D. A store sells two types of gift boxes Regular and
Deluxe. The Regular has 12 apples and 8 bananas, the Deluxe
has 24 apples and 24 bananas. We have 3 boxes of Regular and
4 boxes of Deluxe. How many apples and bananas are there?
48. b. 3(2x – 4) + 2(4 – 5x) expand
= 6x – 12 + 8 – 10x
= –4x – 4
Expressions
Distributive law gives us the option of expanding the content of
the parentheses i.e. extract items out of containers.
Let A stands for apple and B stands for banana,
Example D. A store sells two types of gift boxes Regular and
Deluxe. The Regular has 12 apples and 8 bananas, the Deluxe
has 24 apples and 24 bananas. We have 3 boxes of Regular and
4 boxes of Deluxe. How many apples and bananas are there?
49. b. 3(2x – 4) + 2(4 – 5x) expand
= 6x – 12 + 8 – 10x
= –4x – 4
Expressions
Distributive law gives us the option of expanding the content of
the parentheses i.e. extract items out of containers.
Let A stands for apple and B stands for banana,
then Regular = (12A + 8B) and Deluxe = (24A + 24B).
Example D. A store sells two types of gift boxes Regular and
Deluxe. The Regular has 12 apples and 8 bananas, the Deluxe
has 24 apples and 24 bananas. We have 3 boxes of Regular and
4 boxes of Deluxe. How many apples and bananas are there?
50. b. 3(2x – 4) + 2(4 – 5x) expand
= 6x – 12 + 8 – 10x
= –4x – 4
Expressions
Distributive law gives us the option of expanding the content of
the parentheses i.e. extract items out of containers.
Let A stands for apple and B stands for banana,
then Regular = (12A + 8B) and Deluxe = (24A + 24B).
Three boxes of Regular and four boxes of Deluxe is
3(12A + 8B) + 4(24A + 24B)
Example D. A store sells two types of gift boxes Regular and
Deluxe. The Regular has 12 apples and 8 bananas, the Deluxe
has 24 apples and 24 bananas. We have 3 boxes of Regular and
4 boxes of Deluxe. How many apples and bananas are there?
51. b. 3(2x – 4) + 2(4 – 5x) expand
= 6x – 12 + 8 – 10x
= –4x – 4
Expressions
Distributive law gives us the option of expanding the content of
the parentheses i.e. extract items out of containers.
Let A stands for apple and B stands for banana,
then Regular = (12A + 8B) and Deluxe = (24A + 24B).
Three boxes of Regular and four boxes of Deluxe is
3(12A + 8B) + 4(24A + 24B)
= 36A + 24B + 96A + 96B = 132A + 120B
Example D. A store sells two types of gift boxes Regular and
Deluxe. The Regular has 12 apples and 8 bananas, the Deluxe
has 24 apples and 24 bananas. We have 3 boxes of Regular and
4 boxes of Deluxe. How many apples and bananas are there?
52. b. 3(2x – 4) + 2(4 – 5x) expand
= 6x – 12 + 8 – 10x
= –4x – 4
Expressions
Distributive law gives us the option of expanding the content of
the parentheses i.e. extract items out of containers.
Let A stands for apple and B stands for banana,
then Regular = (12A + 8B) and Deluxe = (24A + 24B).
Three boxes of Regular and four boxes of Deluxe is
3(12A + 8B) + 4(24A + 24B)
= 36A + 24B + 96A + 96B = 132A + 120B
Hence we have 132 apples and 120 bananas.
Example D. A store sells two types of gift boxes Regular and
Deluxe. The Regular has 12 apples and 8 bananas, the Deluxe
has 24 apples and 24 bananas. We have 3 boxes of Regular and
4 boxes of Deluxe. How many apples and bananas are there?
53. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
54. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers.
55. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
56. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
57. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
Example E. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4}
58. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
Example E. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
59. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
Example E. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
60. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
Example E. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
61. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
= –3{–3x – [17 + 8x] – 4}
Example E. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
62. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
= –3{–3x – [17 + 8x] – 4}
Example E. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
63. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
= –3{–3x – [17 + 8x] – 4}
Example E. Expand.
= –3{– 3x – 17 – 8x – 4}
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
64. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
= –3{–3x – [17 + 8x] – 4}
Example E. Expand.
= –3{– 3x – 17 – 8x – 4}
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
simplify,
65. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
= –3{–3x – [17 + 8x] – 4}
Example E. Expand.
= –3{– 3x – 17 – 8x – 4}
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
simplify,
= –3{–11x – 21}
66. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
In mathematics, ( )’s, [ ]’s, and { }’s are often used to
distinguish the layers. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
Always simplify the content of a set of parentheses first before
expanding it.
= –3{–3x – [5 + 8x + 12] – 4}
= –3{–3x – [17 + 8x] – 4}
Example E. Expand.
= –3{– 3x – 17 – 8x – 4}
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
simplify,
= –3{–11x – 21} expand,
= 33x + 63
69. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q*
70. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Hence is the same as .2x
3
2
3
x
*
71. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
Hence is the same as .2x
3
2
3
x
*
72. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6)
73. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6)
74. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
75. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
76. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
Example F. Combine
4
3
x + 5
4
x
77. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
4
3
x + 5
4
x
Example F. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
78. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
= ( )12 / 12
4
3
x + 5
4
x
4
3
x + 5
4
x
Example F. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
79. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
= ( )12 / 12
4
3
x + 5
4
x
4
3
x + 5
4
x expand and cancel the denominators,
Example F. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
80. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
4
= ( )12 / 12
4
3
x + 5
4
x
4
3
x + 5
4
x expand and cancel the denominators,
Example F. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
81. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
4 3
= ( )12 / 12
4
3
x + 5
4
x
4
3
x + 5
4
x expand and cancel the denominators,
Example F. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
82. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
4 3
= ( )12 / 12
4
3
x + 5
4
x
4
3
x + 5
4
x expand and cancel the denominators,
= (4*4 x + 5*3x) / 12
Example F. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
83. Expressions
X-terms with fractional coefficients may be written in two ways,
p
q
x or
px
q
( but not as ) since
p
qx
p
q x =
p
q
x
1
= px
q
Example E. Evaluate if x = 6.4
3
x
we getSet x = (6) in
4
3
2
Hence is the same as .2x
3
2
3
x
*
4
3
x, (6) = 8
We may use the multiplier method to combine fraction terms
i.e. multiply the problem by the LCD and divide by the LCD.
31x
12
4 3
= ( )12 / 12
4
3
x + 5
4
x
4
3
x + 5
4
x expand and cancel the denominators,
= (4*4 x + 5*3x) / 12 = (16x + 15x) /12 =
Example F. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
85. Expressions
C. Starting from the innermost ( ) expand and simplify.
29. x + 2[6 + 4(–3 + 5x)] 30. –5[ x – 4(–7 – 5x)] + 6
31. 8 – 2[4(–3x + 5) + 6x] + x 32. –14x + 5[x – 4(–5x + 15)]
33. –7x + 3{8 – [6(x – 2) –3] – 5x}
34. –3{8 – [6(x – 2) –3] – 5x} – 5[x – 3(–5x + 4)]
35. 4[5(3 – 2x) – 6x] – 3{x – 2[x – 3(–5x + 4)]}
2
3
x + 3
4
x36.
4
3
x – 3
4
x37.
3
8
x – 5
6
x39.5
8
x + 1
6
x38. – –
D. Combine using the LCD-multiplication method
40. Do 36 – 39 by the cross–multiplication method.
86. Expressions
42. As in example D with gift boxes Regular and Deluxe, the
Regular contains12 apples and 8 bananas, the Deluxe has 24
apples and 24 bananas. For large orders we may ship them in
crates or freight-containers where a crate contains 100 boxes
Regular and 80 boxes Deluxe and a container holds 150
Regular boxes and 100 Deluxe boxes.
King Kong ordered 4 crates and 5 containers, how many of
each type of fruit does King Kong have?
41. As in example D with gift boxes Regular and Deluxe, the
Regular contains12 apples and 8 bananas, the Deluxe has 24
apples and 24 bananas. Joe has 6 Regular boxes and 8
Deluxe boxes. How many of each type of fruit does he have?