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2 5 rational equations word-problems
The document discusses using rational equations to solve word problems involving costs shared among groups of people. It provides an example where a taxi costs $20 to rent for a group of x people, with the cost shared equally. If one person leaves the group, the remaining people each pay $1 more. Setting up the cost equations and subtracting them allows determining that x must equal 5 people for the equations to hold true. A table is suggested to organize calculations for different inputs when solving similar rational equation word problems.
4 multiplication and division of rational expressions
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the top expressions divided by the product of the bottom expressions. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
2 the least common multiple and clearing the denominators
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required across different college applications in each subject area.
5.1 hw sequences and summation notation x
1. The document provides examples of infinite series that converge to a finite sum. It gives the series 1/2 + 1/4 + 1/8 + 1/16 + ..., which represents taking half of the remaining amount repeatedly, and shows that it converges to 1.
2. It asks the reader to determine the sums of several other infinite series using similar reasoning:
- The series 1/3 + 1/9 + 1/27 + 1/81 + ... is shown to equal 1 by factoring out 1/3 from each term.
- Factoring out 1/4 from the terms shows the series 1/4 + 1/16 + 1/64 + 1/256 + ... equals
5 4 equations that may be reduced to quadratics-x
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting expressions like (x/(x-1)) with a variable y, solving the resulting quadratic equation for y, and then substituting back to find values of x. This process of solving two simpler equations through substitution is demonstrated to solve equations that are otherwise difficult to solve directly.
5 3 the graphs of quadratic equations-x
The document discusses graphs of quadratic equations. It explains that quadratic equations form parabolic graphs rather than straight lines. It provides examples of graphing quadratic functions by first finding the vertex using a formula, then making a table of x and y values centered around the vertex to plot points symmetrically. Key properties of parabolas are that they are symmetric around the vertex, which is the highest/lowest point on the center line.
5 2 solving 2nd degree equations-x
The document discusses three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It is only applicable if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
5 1 complex numbers-x
1) Complex numbers are numbers of the form a + bi, where a and b are real numbers. a is called the real part and bi is called the imaginary part.
2) To add or subtract complex numbers, treat i as a variable and combine like terms.
3) To multiply complex numbers, use FOIL and set i^2 equal to -1 to simplify the result.
Recommended
2 5 rational equations word-problems
The document discusses using rational equations to solve word problems involving costs shared among groups of people. It provides an example where a taxi costs $20 to rent for a group of x people, with the cost shared equally. If one person leaves the group, the remaining people each pay $1 more. Setting up the cost equations and subtracting them allows determining that x must equal 5 people for the equations to hold true. A table is suggested to organize calculations for different inputs when solving similar rational equation word problems.
4 multiplication and division of rational expressions
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which states that the product of two rational expressions is equal to the product of the top expressions divided by the product of the bottom expressions. Examples are provided to demonstrate simplifying rational expression products by factoring and canceling like terms.
2 the least common multiple and clearing the denominators
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required across different college applications in each subject area.
5.1 hw sequences and summation notation x
1. The document provides examples of infinite series that converge to a finite sum. It gives the series 1/2 + 1/4 + 1/8 + 1/16 + ..., which represents taking half of the remaining amount repeatedly, and shows that it converges to 1.
2. It asks the reader to determine the sums of several other infinite series using similar reasoning:
- The series 1/3 + 1/9 + 1/27 + 1/81 + ... is shown to equal 1 by factoring out 1/3 from each term.
- Factoring out 1/4 from the terms shows the series 1/4 + 1/16 + 1/64 + 1/256 + ... equals
5 4 equations that may be reduced to quadratics-x
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting expressions like (x/(x-1)) with a variable y, solving the resulting quadratic equation for y, and then substituting back to find values of x. This process of solving two simpler equations through substitution is demonstrated to solve equations that are otherwise difficult to solve directly.
5 3 the graphs of quadratic equations-x
The document discusses graphs of quadratic equations. It explains that quadratic equations form parabolic graphs rather than straight lines. It provides examples of graphing quadratic functions by first finding the vertex using a formula, then making a table of x and y values centered around the vertex to plot points symmetrically. Key properties of parabolas are that they are symmetric around the vertex, which is the highest/lowest point on the center line.
5 2 solving 2nd degree equations-x
The document discusses three methods for solving second degree equations (ax2 + bx + c = 0):
1) The square-root method, which is used when the x-term is missing. It involves solving for x2 and taking the square root to find x.
2) Factoring, which involves factoring the equation into the form (ax + b)(cx + d) = 0. It is only applicable if b2 - 4ac is a perfect square.
3) The quadratic formula, which can be used to solve any second degree equation.
5 1 complex numbers-x
1) Complex numbers are numbers of the form a + bi, where a and b are real numbers. a is called the real part and bi is called the imaginary part.
2) To add or subtract complex numbers, treat i as a variable and combine like terms.
3) To multiply complex numbers, use FOIL and set i^2 equal to -1 to simplify the result.
4 6 radical equations-x
Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.
4 5 fractional exponents-x
The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.
4 4 more on algebra of radicals-x
The document discusses algebra of radicals. It provides rules for simplifying expressions involving radicals, such as √x·y = √x·√y and √x·√x = x. An example problem is worked through step-by-step, simplifying the expression 3√3 * √2* 2 * √2 * √3 * √2. The concept of conjugates is also introduced, where the conjugate of x + y is x - y.
4 3 algebra of radicals-x
The document discusses rules for simplifying expressions involving radicals. It presents the multiplication rule that √x∙y = √x∙√y and the division rule. It then gives examples of simplifying expressions such as √3∙√3 = 3, 3√3∙√3 = 9, and (3√3)2 = 27 using these rules.
4 2 rules of radicals-x
The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a perfect square is the number itself, and that the square root of a product is the product of the individual square roots. Examples are provided to demonstrate simplifying radical expressions by extracting square roots from the radicand in steps using these rules.
4 1 radicals and pythagorean theorem-x
The document discusses square roots and radicals. It defines the square root operation as finding the number that, when squared, equals the given number. It provides a table of common square numbers and their square roots that should be memorized. It also describes how to estimate the square root of numbers between values in the table by interpolating between the two closest square roots. A scientific calculator is needed to evaluate more complex square roots.
3 6 2 d linear inequalities-x
The document discusses solving linear inequalities in two variables (x and y). It explains that the solutions to inequalities in x are segments of the real line, while the solutions to inequalities in both x and y are regions of the plane. It then provides an example of using the graph of y=x to identify the regions defined by y>x and y<x. Finally, it discusses the general process of solving linear inequalities Ax + By > C or Ax + By < C by graphing the line Ax + By = C and using point testing to determine which half-plane satisfies the given inequality.
3 5 rectangular system and lines-x
The document describes the rectangular coordinate system. It defines a coordinate system as assigning positions in a plane or space with addresses. The rectangular coordinate system uses a grid with two perpendicular axes (x and y) intersecting at the origin (0,0). Any point in the plane is located by its coordinates (x,y), where x is the distance right or left of the origin and y is the distance up or down. The four quadrants divided by the axes are labeled based on the signs of the x and y coordinates.
3 4 absolute inequalities-algebraic-x
The document discusses absolute value inequalities and provides examples of how to represent them graphically. It explains that an inequality of the form |x| < c can be rewritten as -c < x < c, representing all values within c units of 0. An inequality of the form |x| > c is split into two inequalities x < -c or c < x, representing values more than c units from 0. Examples are given of drawing the solutions to |x| < 7, |x| > 7, and solving |3 - 2x| < 7 algebraically then graphically.
3 3 absolute inequalities-geom-x
The document discusses solving absolute value inequalities using a geometric method. It introduces absolute value inequalities as statements about distances on the real number line. Example A explains that |x| < 7 represents all numbers within 7 units of 0, or between -7 and 7. Example B translates |x - 2| < 3 to mean the distance between x and 2 must be less than 3, with the solution being -1 < x < 5. The document outlines rules for one-piece and two-piece absolute value inequalities and works through additional examples.
3 2 absolute value equations-x
The document discusses the definition and properties of absolute value equations. It defines absolute value as the distance from a number to zero on the real number line. It presents rules for solving absolute value equations, including rewriting an absolute value equation as two separate equations without the absolute value signs, and the property that if the absolute value of an expression is equal to a positive number a, then the expression must equal -a or a. Examples are provided to demonstrate solving absolute value equations using these rules and properties.
3 1 the real line and linear inequalities-x
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on the real number line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the positions of two numbers on the real number line, with the number farther to the right said to be greater than the number to its left. The document provides examples of inequalities and how to represent sets of numbers using inequalities, such as all numbers between two values a and b. It also outlines steps for solving inequalities algebraically.
2 8 variations-xy
The document discusses direct and inverse variations. It defines direct variation as y=kx, where k is a constant, and inverse variation as y=k/x. Examples are given of translating phrases describing variations into equations. For a direct variation problem between variables y and x where y=-4 when x=-6, the specific equation is found to be y=2/3x. For an inverse variation between weight W and distance D from Earth's center, the person's weight 6000 miles above the surface is calculated using the general inverse variation equation W=k/D^2.
4 5 fractional exponents-x
This document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples like 43 = 64. Rules for exponents are covered, including:
- Multiply-Add Rule: ANAK = AN+K
- Divide-Subtract Rule: AN/AK = AN-K
- Negative exponents represent reciprocals, so A-K = 1/AK
Fractional exponents are introduced, along with the 0-Power Rule that A0 = 1.
1 2 2nd-degree equation and word problems-x
This document discusses solving polynomial equations by factoring. It provides an example of solving the equation x^2 - 2x - 3 = 0 by: 1) factoring the trinomial to get (x - 3)(x + 1) = 0, 2) setting each factor equal to 0 to get x - 3 = 0 and x + 1 = 0, and 3) extracting the solutions x = 3 and x = -1. It then works through another example of solving 2 = 2x^2 - 3x by similar factoring steps to get the solutions x = -1/2, x = 2. Finally, it introduces word problems that can be modeled by quadratic equations of the form AB = C.
1 4 homework
This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
5 4 equations that may be reduced to quadratics
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting (x/(x-1)) with y, then solving the resulting quadratic equation for y and back substituting to find values of x. This reduction technique converts difficult equations into two easier equations to solve: the quadratic after substitution and then solving for the original variable.
5 3 the graphs of quadratic equations
- The document discusses graphs of linear and quadratic equations.
- Linear equations produce straight line graphs, while quadratic equations produce curved graphs called parabolas.
- To graph a parabola, one finds the vertex using the vertex formula, then makes a table of x and y values centered around the vertex to plot the points symmetrically.
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4 6 radical equations-x
Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.
4 5 fractional exponents-x
The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.
4 4 more on algebra of radicals-x
The document discusses algebra of radicals. It provides rules for simplifying expressions involving radicals, such as √x·y = √x·√y and √x·√x = x. An example problem is worked through step-by-step, simplifying the expression 3√3 * √2* 2 * √2 * √3 * √2. The concept of conjugates is also introduced, where the conjugate of x + y is x - y.
4 3 algebra of radicals-x
The document discusses rules for simplifying expressions involving radicals. It presents the multiplication rule that √x∙y = √x∙√y and the division rule. It then gives examples of simplifying expressions such as √3∙√3 = 3, 3√3∙√3 = 9, and (3√3)2 = 27 using these rules.
4 2 rules of radicals-x
The document discusses rules for simplifying radical expressions. It states the square root and multiplication rules, which are that the square root of a perfect square is the number itself, and that the square root of a product is the product of the individual square roots. Examples are provided to demonstrate simplifying radical expressions by extracting square roots from the radicand in steps using these rules.
4 1 radicals and pythagorean theorem-x
The document discusses square roots and radicals. It defines the square root operation as finding the number that, when squared, equals the given number. It provides a table of common square numbers and their square roots that should be memorized. It also describes how to estimate the square root of numbers between values in the table by interpolating between the two closest square roots. A scientific calculator is needed to evaluate more complex square roots.
3 6 2 d linear inequalities-x
The document discusses solving linear inequalities in two variables (x and y). It explains that the solutions to inequalities in x are segments of the real line, while the solutions to inequalities in both x and y are regions of the plane. It then provides an example of using the graph of y=x to identify the regions defined by y>x and y<x. Finally, it discusses the general process of solving linear inequalities Ax + By > C or Ax + By < C by graphing the line Ax + By = C and using point testing to determine which half-plane satisfies the given inequality.
3 5 rectangular system and lines-x
The document describes the rectangular coordinate system. It defines a coordinate system as assigning positions in a plane or space with addresses. The rectangular coordinate system uses a grid with two perpendicular axes (x and y) intersecting at the origin (0,0). Any point in the plane is located by its coordinates (x,y), where x is the distance right or left of the origin and y is the distance up or down. The four quadrants divided by the axes are labeled based on the signs of the x and y coordinates.
3 4 absolute inequalities-algebraic-x
The document discusses absolute value inequalities and provides examples of how to represent them graphically. It explains that an inequality of the form |x| < c can be rewritten as -c < x < c, representing all values within c units of 0. An inequality of the form |x| > c is split into two inequalities x < -c or c < x, representing values more than c units from 0. Examples are given of drawing the solutions to |x| < 7, |x| > 7, and solving |3 - 2x| < 7 algebraically then graphically.
3 3 absolute inequalities-geom-x
The document discusses solving absolute value inequalities using a geometric method. It introduces absolute value inequalities as statements about distances on the real number line. Example A explains that |x| < 7 represents all numbers within 7 units of 0, or between -7 and 7. Example B translates |x - 2| < 3 to mean the distance between x and 2 must be less than 3, with the solution being -1 < x < 5. The document outlines rules for one-piece and two-piece absolute value inequalities and works through additional examples.
3 2 absolute value equations-x
The document discusses the definition and properties of absolute value equations. It defines absolute value as the distance from a number to zero on the real number line. It presents rules for solving absolute value equations, including rewriting an absolute value equation as two separate equations without the absolute value signs, and the property that if the absolute value of an expression is equal to a positive number a, then the expression must equal -a or a. Examples are provided to demonstrate solving absolute value equations using these rules and properties.
3 1 the real line and linear inequalities-x
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on the real number line, with positive numbers to the right of zero and negative numbers to the left. An inequality relates the positions of two numbers on the real number line, with the number farther to the right said to be greater than the number to its left. The document provides examples of inequalities and how to represent sets of numbers using inequalities, such as all numbers between two values a and b. It also outlines steps for solving inequalities algebraically.
2 8 variations-xy
The document discusses direct and inverse variations. It defines direct variation as y=kx, where k is a constant, and inverse variation as y=k/x. Examples are given of translating phrases describing variations into equations. For a direct variation problem between variables y and x where y=-4 when x=-6, the specific equation is found to be y=2/3x. For an inverse variation between weight W and distance D from Earth's center, the person's weight 6000 miles above the surface is calculated using the general inverse variation equation W=k/D^2.
4 5 fractional exponents-x
This document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples like 43 = 64. Rules for exponents are covered, including:
- Multiply-Add Rule: ANAK = AN+K
- Divide-Subtract Rule: AN/AK = AN-K
- Negative exponents represent reciprocals, so A-K = 1/AK
Fractional exponents are introduced, along with the 0-Power Rule that A0 = 1.
1 2 2nd-degree equation and word problems-x
This document discusses solving polynomial equations by factoring. It provides an example of solving the equation x^2 - 2x - 3 = 0 by: 1) factoring the trinomial to get (x - 3)(x + 1) = 0, 2) setting each factor equal to 0 to get x - 3 = 0 and x + 1 = 0, and 3) extracting the solutions x = 3 and x = -1. It then works through another example of solving 2 = 2x^2 - 3x by similar factoring steps to get the solutions x = -1/2, x = 2. Finally, it introduces word problems that can be modeled by quadratic equations of the form AB = C.
1 4 homework
This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
5 4 equations that may be reduced to quadratics
The document discusses reducing equations to quadratic equations using substitution. It explains that if a pattern is repeated in an expression, a variable can be substituted to simplify the equation. Examples show substituting (x/(x-1)) with y, then solving the resulting quadratic equation for y and back substituting to find values of x. This reduction technique converts difficult equations into two easier equations to solve: the quadratic after substitution and then solving for the original variable.
5 3 the graphs of quadratic equations
- The document discusses graphs of linear and quadratic equations.
- Linear equations produce straight line graphs, while quadratic equations produce curved graphs called parabolas.
- To graph a parabola, one finds the vertex using the vertex formula, then makes a table of x and y values centered around the vertex to plot the points symmetrically.