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.
1) The document defines terms, constants, coefficients, and discusses how to solve one-step and two-step equations. It provides examples of solving equations involving addition, subtraction, multiplication and division.
2) Sample one-step equations are provided along with the steps to solve each type. Equations involving addition, subtraction, multiplication and division are worked through as examples.
3) A two-step practice problem is given along with the answers. Solving two-step equations is introduced.
1) The document discusses various forms of equations for lines, including slope-intercept form, standard form, and point-slope form. It provides definitions and examples of writing equations of lines given the slope and y-intercept or given two points on the line.
2) Key concepts covered include writing the equation of a line given its slope m and y-intercept b using slope-intercept form y=mx+b, or given slope m and a point (x1,y1) using point-slope form y-y1=m(x-x1).
3) Examples are provided for writing equations of lines using slope-intercept form when given slope and y-intercept, and using point-
This document discusses inequalities and their graphs. It defines an inequality as a statement that two expressions are not equal. It explains the symbols used in inequalities like <, >, ≤, and ≥ and how to graph them correctly, with open or closed circles depending on if it's a < or ≤. Examples are provided of writing and graphing simple one-variable inequalities and having the reader practice graphing some on their own. Key points covered include understanding the direction the solution goes based on what side the variable is on and coloring in or leaving open the circle depending on the symbol used.
This document provides instructions for solving systems of equations using elimination. It demonstrates eliminating variables by adding or subtracting equations. Sample systems are worked through, showing the steps of identifying which variable to eliminate, combining the equations accordingly, solving for one variable, then substituting back into the original equations to solve for the other. The solutions are checked in both equations to verify they satisfy the system.
IT'S A PRESENTATION ON QUADRATIC EQUATION PART 1, CLASS 10, CHAPTER 4, IT STARTS WITH THE SHAPE PARABOLA AND IT'S DAY TO DAY LIFE EXAMPLES, AS WE PROCEED FURTHER WE SOLVE SOME EXPRESSIONS, WE COVERT IT INTO QUADRATIC EQUATIONS. AFTERWARDS, WE LEARN HOW TO FORM STANDARD QUADRATIC EQUATIONS WITH EXAMPLES (WORD PROBLEMS).
The document discusses proportional relationships in triangles using theorems about parallel lines and angle bisectors. It provides examples of applying the side-splitter theorem and triangle-angle-bisector theorem to find unknown values in various geometric figures by setting up proportional relationships between corresponding sides or segments. Readers are given practice problems applying these proportional relationship theorems to find specific variable values in diagrams.
A linear inequality is similar to a linear equation but uses inequality symbols like < or > instead of =. A solution to a linear inequality is any coordinate pair that makes the inequality true. A linear inequality describes a half-plane region on a coordinate plane where all points in the region satisfy the inequality, with the boundary line given by the related equation. To graph a linear inequality, you solve it for y, graph the boundary line as solid or dotted, and shade the correct half-plane above or below the line.
This document discusses solving two-step equations. It provides examples of solving equations through undoing operations in reverse order based on the PEMDAS method. Students are shown how to solve equations, check their work, and apply equation solving to real-life word problems involving variables like the number of pickles a soccer player can eat or the price of plants at a fundraiser.
1) The document defines terms, constants, coefficients, and discusses how to solve one-step and two-step equations. It provides examples of solving equations involving addition, subtraction, multiplication and division.
2) Sample one-step equations are provided along with the steps to solve each type. Equations involving addition, subtraction, multiplication and division are worked through as examples.
3) A two-step practice problem is given along with the answers. Solving two-step equations is introduced.
1) The document discusses various forms of equations for lines, including slope-intercept form, standard form, and point-slope form. It provides definitions and examples of writing equations of lines given the slope and y-intercept or given two points on the line.
2) Key concepts covered include writing the equation of a line given its slope m and y-intercept b using slope-intercept form y=mx+b, or given slope m and a point (x1,y1) using point-slope form y-y1=m(x-x1).
3) Examples are provided for writing equations of lines using slope-intercept form when given slope and y-intercept, and using point-
This document discusses inequalities and their graphs. It defines an inequality as a statement that two expressions are not equal. It explains the symbols used in inequalities like <, >, ≤, and ≥ and how to graph them correctly, with open or closed circles depending on if it's a < or ≤. Examples are provided of writing and graphing simple one-variable inequalities and having the reader practice graphing some on their own. Key points covered include understanding the direction the solution goes based on what side the variable is on and coloring in or leaving open the circle depending on the symbol used.
This document provides instructions for solving systems of equations using elimination. It demonstrates eliminating variables by adding or subtracting equations. Sample systems are worked through, showing the steps of identifying which variable to eliminate, combining the equations accordingly, solving for one variable, then substituting back into the original equations to solve for the other. The solutions are checked in both equations to verify they satisfy the system.
IT'S A PRESENTATION ON QUADRATIC EQUATION PART 1, CLASS 10, CHAPTER 4, IT STARTS WITH THE SHAPE PARABOLA AND IT'S DAY TO DAY LIFE EXAMPLES, AS WE PROCEED FURTHER WE SOLVE SOME EXPRESSIONS, WE COVERT IT INTO QUADRATIC EQUATIONS. AFTERWARDS, WE LEARN HOW TO FORM STANDARD QUADRATIC EQUATIONS WITH EXAMPLES (WORD PROBLEMS).
The document discusses proportional relationships in triangles using theorems about parallel lines and angle bisectors. It provides examples of applying the side-splitter theorem and triangle-angle-bisector theorem to find unknown values in various geometric figures by setting up proportional relationships between corresponding sides or segments. Readers are given practice problems applying these proportional relationship theorems to find specific variable values in diagrams.
A linear inequality is similar to a linear equation but uses inequality symbols like < or > instead of =. A solution to a linear inequality is any coordinate pair that makes the inequality true. A linear inequality describes a half-plane region on a coordinate plane where all points in the region satisfy the inequality, with the boundary line given by the related equation. To graph a linear inequality, you solve it for y, graph the boundary line as solid or dotted, and shade the correct half-plane above or below the line.
This document discusses solving two-step equations. It provides examples of solving equations through undoing operations in reverse order based on the PEMDAS method. Students are shown how to solve equations, check their work, and apply equation solving to real-life word problems involving variables like the number of pickles a soccer player can eat or the price of plants at a fundraiser.
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
This document provides an overview of graphing linear equations. It defines key terms like solutions, intercepts, and linear models. Examples are given to show how to graph equations by finding intercepts or using a table of points. Horizontal and vertical lines are discussed as special cases of linear equations. The document concludes with an example of using a linear equation to model a real-world situation involving monthly phone costs.
This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It was named after the Greek mathematician Pythagoras, who lived in the 6th century BC. The theorem can be used to calculate unknown side lengths in right triangles. Some examples are also given to demonstrate applying the theorem.
This document provides an overview of linear functions and equations. It defines linear equations as having the standard form Ax + By = C, with examples and how to identify linear vs. nonlinear equations. Linear functions are defined as having the form f(x) = mx + b. The document discusses slope, x-intercepts, y-intercepts, and how to graph linear equations from these components. It also covers representing linear functions in slope-intercept form as y = mx + b, and point-slope form as y - y1 = m(x - x1).
The document defines key polynomial vocabulary including:
- Terms are numbers or products of numbers and variables raised to powers. Coefficients are numerical factors of terms. Constants are terms that are only numbers.
- Polynomials are sums of terms involving variables raised to whole number exponents, with no variables in denominators.
- Types of polynomials include monomials (1 term), binomials (2 terms), and trinomials (3 terms). Degree is the largest exponent of any term.
- Operations on polynomials include adding/subtracting like terms, multiplying using distribution and FOIL, dividing using long division, and special products like (a+b)2 and (a+b)(a
The document contains examples of solving various types of algebraic equations including:
1) Equations with multiplication and subtraction or addition such as 2x - 4 = 8 and 5x + 10 = 80.
2) Equations with fractions such as 2/3x + 2 = 8.
3) Equations involving division such as x/5 + 2 = 8.
4) Equations with collecting like terms such as 4x + 6x + 20 = 80.
5) Equations using the distributive property such as 10x – 3x -12 = 4x – 9x + 48.
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
1. The document provides instructions for simplifying square root expressions using properties like the product and quotient properties. It also covers adding, subtracting, and rationalizing denominators of square root expressions.
2. Examples are given to practice simplifying expressions, adding/subtracting roots, and rationalizing denominators. A word problem asks students to find the width of a square poster using its given area.
3. A quiz is outlined to assess understanding of simplifying roots, rationalizing denominators, and adding/subtracting roots. Students are instructed to complete worksheets and ask any remaining questions.
This document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a ≠ 0.
2) It discusses the importance of quadratic equations, noting that the term "quadratic" comes from the variable being squared (x2) and that a quadratic equation is a trinomial expression with three terms.
3) It presents the method of factorization to solve quadratic equations, showing that if ax2 + bx + c = (rx + p)(sx + q) = 0, then the solutions are x1 = -p/r and x2 = -q/s
3 2 solving systems of equations (elimination method)Hazel Joy Chong
The document describes the elimination method for solving systems of equations. The key steps are:
1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It defines the key terms hypotenuse, legs, and explains the a^2 + b^2 = c^2 formula. Examples are given to demonstrate applying the theorem to find the unknown side of a right triangle given the other two sides. Practice problems are provided to reinforce the concept.
The document introduces some key terms used in algebra. It defines an algebraic term as having a numerical coefficient and one or more literal coefficients. Like terms are terms that have the same literal coefficient, while unlike terms have different literal coefficients. Algebraic expressions are formed by combining algebraic terms with addition or subtraction. Expressions can be monomial, binomial, trinomial or polynomial depending on the number of terms. Finally, an equation is defined as a statement that sets the left-hand side equal to the right-hand side.
The document provides steps for factoring polynomial expressions:
1) Separate the expression into terms with common factors and group those terms;
2) Find the greatest common factor (GCF) of the grouped terms;
3) Enclose the GCF and the grouped terms in parentheses to arrive at the factored expression.
Worked examples demonstrate factoring various polynomials, including using negative signs properly.
This document discusses using the point-slope form to find the equation of a line given a slope and point. It provides the point-slope form equation, and examples of finding the line equation for different slopes and points. Exercises are provided for the reader to practice finding additional line equations using given slopes and points.
Solving System of Equations by SubstitutionTwinkiebear7
1) The document discusses solving systems of equations using substitution. It provides 5 steps for solving a system by substitution: 1) solve one equation for a variable, 2) substitute into the other equation, 3) solve the new equation, 4) plug back in to find the other variable, and 5) check the solution.
2) It then works through examples, showing that substitution is easiest when one equation is already solved for a variable. It also notes that if the final step results in a false statement, there are no solutions, and if true, there are infinitely many solutions.
Algebra is the study of mathematical symbols and rules for calculating those symbols, which allows numbers to be represented by variables. An algebraic expression combines constants and variables using operations like addition, subtraction, multiplication and division. Expressions can be monomials with one term, binomials with two terms, or trinomials with three terms. To multiply algebraic expressions, the signs and coefficients are multiplied, and the variables are multiplied using exponent rules.
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 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.
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
This document provides an overview of graphing linear equations. It defines key terms like solutions, intercepts, and linear models. Examples are given to show how to graph equations by finding intercepts or using a table of points. Horizontal and vertical lines are discussed as special cases of linear equations. The document concludes with an example of using a linear equation to model a real-world situation involving monthly phone costs.
This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It was named after the Greek mathematician Pythagoras, who lived in the 6th century BC. The theorem can be used to calculate unknown side lengths in right triangles. Some examples are also given to demonstrate applying the theorem.
This document provides an overview of linear functions and equations. It defines linear equations as having the standard form Ax + By = C, with examples and how to identify linear vs. nonlinear equations. Linear functions are defined as having the form f(x) = mx + b. The document discusses slope, x-intercepts, y-intercepts, and how to graph linear equations from these components. It also covers representing linear functions in slope-intercept form as y = mx + b, and point-slope form as y - y1 = m(x - x1).
The document defines key polynomial vocabulary including:
- Terms are numbers or products of numbers and variables raised to powers. Coefficients are numerical factors of terms. Constants are terms that are only numbers.
- Polynomials are sums of terms involving variables raised to whole number exponents, with no variables in denominators.
- Types of polynomials include monomials (1 term), binomials (2 terms), and trinomials (3 terms). Degree is the largest exponent of any term.
- Operations on polynomials include adding/subtracting like terms, multiplying using distribution and FOIL, dividing using long division, and special products like (a+b)2 and (a+b)(a
The document contains examples of solving various types of algebraic equations including:
1) Equations with multiplication and subtraction or addition such as 2x - 4 = 8 and 5x + 10 = 80.
2) Equations with fractions such as 2/3x + 2 = 8.
3) Equations involving division such as x/5 + 2 = 8.
4) Equations with collecting like terms such as 4x + 6x + 20 = 80.
5) Equations using the distributive property such as 10x – 3x -12 = 4x – 9x + 48.
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
1. The document provides instructions for simplifying square root expressions using properties like the product and quotient properties. It also covers adding, subtracting, and rationalizing denominators of square root expressions.
2. Examples are given to practice simplifying expressions, adding/subtracting roots, and rationalizing denominators. A word problem asks students to find the width of a square poster using its given area.
3. A quiz is outlined to assess understanding of simplifying roots, rationalizing denominators, and adding/subtracting roots. Students are instructed to complete worksheets and ask any remaining questions.
This document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a ≠ 0.
2) It discusses the importance of quadratic equations, noting that the term "quadratic" comes from the variable being squared (x2) and that a quadratic equation is a trinomial expression with three terms.
3) It presents the method of factorization to solve quadratic equations, showing that if ax2 + bx + c = (rx + p)(sx + q) = 0, then the solutions are x1 = -p/r and x2 = -q/s
3 2 solving systems of equations (elimination method)Hazel Joy Chong
The document describes the elimination method for solving systems of equations. The key steps are:
1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It defines the key terms hypotenuse, legs, and explains the a^2 + b^2 = c^2 formula. Examples are given to demonstrate applying the theorem to find the unknown side of a right triangle given the other two sides. Practice problems are provided to reinforce the concept.
The document introduces some key terms used in algebra. It defines an algebraic term as having a numerical coefficient and one or more literal coefficients. Like terms are terms that have the same literal coefficient, while unlike terms have different literal coefficients. Algebraic expressions are formed by combining algebraic terms with addition or subtraction. Expressions can be monomial, binomial, trinomial or polynomial depending on the number of terms. Finally, an equation is defined as a statement that sets the left-hand side equal to the right-hand side.
The document provides steps for factoring polynomial expressions:
1) Separate the expression into terms with common factors and group those terms;
2) Find the greatest common factor (GCF) of the grouped terms;
3) Enclose the GCF and the grouped terms in parentheses to arrive at the factored expression.
Worked examples demonstrate factoring various polynomials, including using negative signs properly.
This document discusses using the point-slope form to find the equation of a line given a slope and point. It provides the point-slope form equation, and examples of finding the line equation for different slopes and points. Exercises are provided for the reader to practice finding additional line equations using given slopes and points.
Solving System of Equations by SubstitutionTwinkiebear7
1) The document discusses solving systems of equations using substitution. It provides 5 steps for solving a system by substitution: 1) solve one equation for a variable, 2) substitute into the other equation, 3) solve the new equation, 4) plug back in to find the other variable, and 5) check the solution.
2) It then works through examples, showing that substitution is easiest when one equation is already solved for a variable. It also notes that if the final step results in a false statement, there are no solutions, and if true, there are infinitely many solutions.
Algebra is the study of mathematical symbols and rules for calculating those symbols, which allows numbers to be represented by variables. An algebraic expression combines constants and variables using operations like addition, subtraction, multiplication and division. Expressions can be monomials with one term, binomials with two terms, or trinomials with three terms. To multiply algebraic expressions, the signs and coefficients are multiplied, and the variables are multiplied using exponent rules.
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 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 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 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 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 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 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 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 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 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.
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 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 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.
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 different methods for solving equations, including:
- Solving 1st and 2nd degree polynomial equations by setting them equal to 0 and using factoring or the quadratic formula.
- Solving rational equations by clearing all denominators using the lowest common denominator.
- Solving equations may require transforming them into polynomial equations first through methods like factoring or factoring by grouping.
Similar to 2 expressions and linear expressions (20)
The document discusses equations of lines. It separates lines into two cases - horizontal/vertical lines which have slope 0 or undefined slope, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their properties like slope and intercept points.
The document discusses the concept of slope of a line. It defines slope as the ratio of the "rise" over the "run" between two points on a line. Specifically:
- The slope of a line is calculated as the change in the y-values (rise) divided by the change in the x-values (run) between two points on the line.
- This formula is easy to memorize and captures the geometric meaning of slope as the tilt of the line.
- An example problem demonstrates calculating the slope of a line between two points by finding the difference in their x- and y-values.
The document describes the rectangular coordinate system. It establishes that a coordinate system assigns positions in a plane using ordered pairs of numbers (x,y). It defines the x-axis, y-axis, and origin at their intersection. Any point is addressed by its coordinates (x,y) where x represents horizontal distance from the origin and y represents vertical distance. The four quadrants divided by the axes are also defined based on positive and negative coordinate values. Reflections of points across the axes and origin are discussed. Finally, it introduces the concept of graphing mathematical relations between x and y coordinates to represent collections of points.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
2 the real line, inequalities and comparative phraseselem-alg-sample
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. Examples are provided of drawing intervals on the number line and solving simple inequalities algebraically. Properties of inequalities like adding the same quantity to both sides preserving the inequality sign are also outlined.
Geometry is the study of shapes, their properties and relationships. Some basic geometric shapes include lines, rays, angles, triangles, quadrilaterals, polygons, circles and three-dimensional shapes like spheres and cubes. Formulas are used to calculate properties of shapes like the area of a triangle is 1/2 * base * height, the circumference of a circle is 2 * pi * radius, and the volume of a cube is side^3.
The document discusses direct and inverse variations. It defines a direct variation as a relationship where y=kx, where k is a constant. An inverse variation is defined as a relationship where y=k/x, where k is a constant. Examples are given of translating phrases describing direct and inverse variations into mathematical equations. The document also explains how to solve word problems involving variations by using given values to find the specific constant k and exact variation equation.
17 applications of proportions and the rational equationselem-alg-sample
The document discusses rational equations word problems involving rates, distances, costs, and number of people. An example problem asks how many people (x) shared a taxi costing $20 if one person leaving causes the remaining people's cost to increase by $1 each. Setting up rational equations and solving leads to the answer that x = 5 people.
16 the multiplier method for simplifying complex fractionselem-alg-sample
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" regular division problem by combining fractions in the numerator and denominator. The second method multiplies the lowest common denominator of all terms to the numerator and denominator to simplify. An example using each method is provided.
15 proportions and the multiplier method for solving rational equationselem-alg-sample
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding or subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate converting rational expressions to equivalent forms with different specified denominators.
14 the lcm and the multiplier method for addition and subtraction of rational...elem-alg-sample
The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
Rational expressions are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions of the form anxn + an-1xn-1 + ... + a1x1 + a0. Rational expressions can be written in either expanded or factored form. The factored form is useful for determining the domain of a rational expression, solving equations involving rational expressions, evaluating inputs, and determining the sign of outputs. The domain of a rational expression excludes values of x that make the denominator equal to 0.
The document discusses applications of factoring polynomials. It provides examples of how factoring can be used to evaluate polynomials by substituting values into the factored form. Factoring is also useful for determining the sign of outputs and for solving polynomial equations, which is described as the most important application of factoring. Examples are given to demonstrate evaluating polynomials both with and without factoring, and checking the answers obtained from factoring using the expanded form.
10 more on factoring trinomials and factoring by formulaselem-alg-sample
The document discusses two methods for factoring trinomials of the form ax^2 + bx + c. The first method is short but not always reliable, while the second method takes more steps but always provides a definite answer. This second method, called the reversed FOIL method, involves finding four numbers that satisfy certain properties to factor the trinomial. An example is worked out step-by-step to demonstrate how to use the reversed FOIL method to factor the trinomial 3x^2 + 5x + 2.
Trinomials are polynomials of the form ax^2 + bx + c, where a, b, and c are numbers. To factor a trinomial, we write it as the product of two binomials (x + u)(x + v) where uv = c and u + v = b. For example, to factor x^2 + 5x + 6, we set uv = 6 and u + v = 5. The only possible values are u = 2 and v = 3, so x^2 + 5x + 6 = (x + 2)(x + 3). Similarly, to factor x^2 - 5x + 6, we set uv = 6 and u + v = -5,
The document discusses factoring quantities by finding common factors. It defines factoring as rewriting a quantity as a product in a nontrivial way. A quantity is prime if it cannot be written as a product other than 1 times the quantity. To factor completely means writing each factor as a product of prime numbers. Examples show finding common factors of quantities, the greatest common factor (GCF), and extracting common factors from sums and differences using the extraction law.
The document discusses methods for multiplying binomial expressions. A binomial is a two-term polynomial of the form ax + b, while a trinomial is a three-term polynomial of the form ax^2 + bx + c. The product of two binomials results in a trinomial. The FOIL method is introduced to multiply binomials, where the Front, Outer, Inner, and Last terms of each binomial are multiplied and combined. Expanding the product of a binomial and a binomial with a leading negative sign requires distributing the negative sign first before using FOIL.
The document discusses polynomial expressions. A polynomial is the sum of monomial terms, where a monomial is a number multiplied by one or more variables raised to a non-negative integer power. Examples show evaluating polynomials by substituting values for variables and calculating each monomial term separately before combining them. A term refers to each monomial in a polynomial. Terms are identified by their variable part, such as the x2-term, x-term, or constant term.
The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then presents and explains the following rules for exponents:
1) Multiplication Rule: ANAK = AN+K
2) Division Rule: AN/AK = AN-K
3) Power Rule: (AN)K = ANK
4) 0-Power Rule: A0 = 1
5) Negative Power Rule: A-K = 1/AK
It provides examples to illustrate how to apply each rule when simplifying expressions with exponents.
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.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
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.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
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 Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
2. Example B.
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 B.
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 B.
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 B.
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 B.
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 B.
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 B.
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 B.
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 B.
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 B.
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 B.
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 B.
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.
16. Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
There are two terms in the linear expression ax + b.
17. Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
There are two terms in the linear expression ax + b.
There are three terms in the expression ax2 + bx + c
18. Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
There are three terms in the expression ax2 + bx + c
19. Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
the x-term the constant term
There are three terms in the expression ax2 + bx + c
20. Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
the x-term the constant term
There are three terms in the expression ax2 + bx + c
the x2-term
21. Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
the x-term the constant term
There are three terms in the expression ax2 + bx + c
the x2-term the constant term
the x-term
22. Just as 3 apples + 5 apples = 8 apples we may combine
3x + 5x = 8x, –3x – 5x = –8x,
Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
the x-term the constant term
There are three terms in the expression ax2 + bx + c
the x2-term the constant term
the x-term
23. Just as 3 apples + 5 apples = 8 apples we may combine
3x + 5x = 8x, –3x – 5x = –8x, 3x – 5x = –2x, and –3x + 5x = 2x.
Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
the x-term the constant term
There are three terms in the expression ax2 + bx + c
the x2-term the constant term
the x-term
24. Just as 3 apples + 5 apples = 8 apples we may combine
3x + 5x = 8x, –3x – 5x = –8x, 3x – 5x = –2x, and –3x + 5x = 2x.
The x-terms can't be combined with the number terms because
they are different type of items just as
2 apple + 3 banana = 2 apple + 3 banana (or 2A + 3B),
i.e. the expression can’t be condensed further.
Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
the x-term the constant term
There are three terms in the expression ax2 + bx + c
the x2-term the constant term
the x-term
25. Just as 3 apples + 5 apples = 8 apples we may combine
3x + 5x = 8x, –3x – 5x = –8x, 3x – 5x = –2x, and –3x + 5x = 2x.
The x-terms can't be combined with the number terms because
they are different type of items just as
2 apple + 3 banana = 2 apple + 3 banana (or 2A + 3B),
i.e. the expression can’t be condensed further.
Hence the expression “2 + 3x” stays as “2 + 3x”, it's not “5x”.
Expressions
Combining Linear Expressions
Every expression is the sum of simpler expressions and each
of these addend(s) is called a term.
Each term is named by its variable-component.
There are two terms in the linear expression ax + b.
the x-term the constant term
There are three terms in the expression ax2 + bx + c
the x2-term the constant term
the x-term
29. For the x-term ax, the number “a” is called the coefficient of
the term.
Expressions
Example C. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
30. 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 C. 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. Hence, 3(5x) = (3*5)x =15x,
Expressions
Example C. 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,
and –2(–4x) = (–2)(–4)x = 8x.
Expressions
Example C. 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.
We may multiply a number with an expression and expand the
result by the distributive law.
Expressions
Example C. 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.
Distributive Law
A(B ± C) = AB ± AC = (B ± C)A
Expressions
Example C. 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 C. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
Example D. Expand then simplify.
a. –5(2x – 4)
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 C. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
Example D. Expand then simplify.
a. –5(2x – 4)
= –5(2x) – (–5)(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 C. Combine.
2x – 4 + 9 – 5x
= 2x – 5x – 4 + 9
= –3x + 5
Example D. Expand then simplify.
a. –5(2x – 4)
= –5(2x) – (–5)(4)
= –10x + 20
45. 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.
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.
Example E. 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?
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.
Let A stands for apple and B stands for banana,
Example E. 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,
then Regular = (12A + 8B) and Deluxe = (24A + 24B).
Example E. 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).
Three boxes of Regular and four boxes of Deluxe is
3(12A + 8B) + 4(24A + 24B)
Example E. 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)
= 36A + 24B + 96A + 96B = 132A + 120B
Example E. 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
Hence we have 132 apples and 120 bananas.
Example E. 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. Expressions
We usually start with the innermost set of parentheses to
simplify an expression that has multiple layers of parentheses.
53. 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.
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. This can't be the case for calculators or
software where [ ] and { } may have other meanings.
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.
Always simplify the content of a set of parentheses first before
expanding it.
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.
Example F. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4}
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 F. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
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.
= –3{–3x – [5 + 8x + 12] – 4}
Example F. 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 F. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
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}
= –3{–3x – [17 + 8x] – 4}
Example F. 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 F. Expand.
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
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 F. Expand.
= –3{– 3x – 17 – 8x – 4}
–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 F. Expand.
= –3{– 3x – 17 – 8x – 4}
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
simplify,
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 F. Expand.
= –3{– 3x – 17 – 8x – 4}
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
simplify,
= –3{–11x – 21}
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 F. Expand.
= –3{– 3x – 17 – 8x – 4}
–3{–3x – [5 – 2(– 4x – 6)] – 4} expand,
simplify,
expand,
simplify,
= –3{–11x – 21} expand,
= 33x + 63
68. 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*
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
Hence is the same as .2x
3
2
3
x
*
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
Example G. Evaluate if x = 6.4
3
x
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 G. 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)
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 G. 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)
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 G. 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
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 G. 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.
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 G. 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 H. Combine
4
3
x + 5
4
x
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 G. 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 H. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. Combine
4
3
x + 5
4
x
Multiply and divide by their LCD =12,
84. 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.
85. 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?