1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.1 Graphs and
Graphing Utilities
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Plot points in the rectangular coordinate system.
Graph equations in the rectangular coordinate system.
Interpret information about a graphing utility’s viewing rectangle or table.
Use a graph to determine intercepts.
Interpret information given by graphs.

3. The Rectangular Coordinate System
We draw a horizontal line
and a vertical line that
intersect at right angles.
The horizontal line is the
x-axis. The vertical line
is the y-axis.
The point of intersection for
these axes is their zero
points, known as the origin.

4. The Rectangular Coordinate System (continued)
Positive numbers are shown
to the right of the origin and
above the origin. Negative
numbers are shown to the
left of the origin and below
the origin.

5. Plotting Points in the Rectangular Coordinate System
Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers,
(x, y).
The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number in each pair, called the y-coordinate, denotes the vertical distance and direction from the origin along the y-axis.

6. Example: Plotting Points in the Rectangular Coordinate System
Plot the point (–2,4)
To plot the point (–2,4),
we move 2 units to the
left of the origin
and 4 units up.

7. Example: Plotting Points in the Rectangular Coordinate System (continued)
Plot the point (4, –2)
To plot the point (4, –2),
we move 4 units to the
right of the origin
and 2 units down.

8. Graphs of Equations
A relationship between two quantities can be expressed as an equation in two variables, such as
A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement.

9. Example: Graphing an Equation Using the Point-Plotting Method
Select integers for x, starting with –4 and ending with 2.

10. Example: Graphing an Equation Using the Point-Plotting Method (continued)
For each value of x, we find the corresponding value for y.

11. Example: Graphing an Equation Using the Point-Plotting Method (continued)
We plot the points
and connect them.

12. Graphing Utilities
Graphing calculators and graphing software packages for computers are referred to as graphing utilities or graphers. To graph an equation in x and y using a graphing utility, enter the equation and specify the size of the viewing rectangle. The size of the viewing rectangle sets minimum and maximum values for both the x-axis and the y-axis. The [–10,10,1] by [–10,10,1] viewing rectangle is called the standard viewing rectangle.

13. Example: Understanding the Viewing Rectangle
What is the meaning of a
[–100,100,50] by [–100,100,10] viewing rectangle?
The minimum x-value is –100.
The maximum x-value is 100.
The distance between
consecutive tic marks on the x-axis is 50.
The minimum y-value is – 100.
The maximum y-value is 100.
The distance between consecutive tic marks on the
y-axis is 10.

14. Intercepts
An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis. The y-coordinate corresponding to an x-intercept is always zero.
A y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. The x-coordinate corresponding to a y-intercept is always zero.

15. Example: Identifying Intercepts
Identify the x- and y-intercepts.
The graph crosses the
x-axis at (–3, 0). Thus,
the x-intercept is –3.
The graph crosses the
y-axis at (0, 5). Thus,
the y-intercept is 5.

16. Example: Interpret Information Given by Graphs
Divorce rates are considerably higher for couples who marry in their teens. The equation models the percentage, d, of marriages that end in divorce after n years if the wife is under 18 at the time of marriage.
Determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage.

17. Example: Interpret Information Given by Graphs (continued)
To determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage we will use the equation
where d is the percentage of marriages that end in divorce after n years.
The percentage of marriages ending in divorce after 15
years when the wife is under 18 at the time of the marriage
is 65%.

18. Example: Interpret Information Given by Graphs (continued)
The graph of d = 4n + 5 is
shown to the left.
We calculated that 65% of
marriages would end in
divorce after 15 years. How
can we check our answer with
the graph?
Our answer is the point
(15, 65) which is on the
graph.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.2 Linear Equations and
Rational Equations
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19

20. Objectives:
Solve linear equations in one variable.
Solve linear equations containing fractions.
Solve rational equations with variables in the denominators.
Recognize identities, conditional equations, and inconsistent equations.
Solve applied problems using mathematical models.

21. Definition of a Linear Equation
A linear equation in one variable x is an equation that can be written in the form
where a and b are real numbers, and

22. Generating Equivalent Equations
An equation can be transformed into an equivalent equation by one or more of the following operations:
1. Simplify an expression by removing grouping symbols and combining like terms.
2. Add (or subtract) the same real number or variable expression on both sides of the equation.
3. Multiply (or divide) by the same nonzero quantity on both sides of the equation.
4. Interchange the two sides of the equation.

23. Solving a Linear Equation
1. Simplify the algebraic expression on each side by removing grouping symbols and combining like terms.
2. Collect all the variable terms on one side and all the numbers, or constant terms, on the other side.
3. Isolate the variable and solve.
4. Check the proposed solution in the original equation.

24. Example: Solving a Linear Equation
Solve and check the linear equation:
Check:

25. Example: Solving a Linear Equation Involving Fractions
Solve and check:
The LCD is 28, we will multiply both sides of the
equation by 28.

26. Example: Solving a Linear Equation Involving Fractions (continued)
Check:

27. Solving Rational Equations
When we solved the equation
we were solving a linear equation with constants in the denominators. A rational equation includes at least one variable in the denominator. For our next example, we will solve the rational equation

28. Example: Solving a Rational Equation
We check for restrictions on the variable
by setting each denominator equal to zero.

29. Example: Solving a Rational Equation (continued)
The LCD is 18x, we will multiply both sides of the
equation by 18x.

30. Example: Solving a Rational Equation to Determine When Two Equations are Equal
Find all values of x for which y1 = y2
We begin by finding the restricted values.
The restricted values are x = – 4 and x = 4

31. Example: Solving a Rational Equation to Determine When Two Equations are Equal (continued)
We multiply both sides of the equation by the LCD

32. Example: Solving a Rational Equation to Determine When Two Equations are Equal (continued)
11 is not a restricted value. Therefore, x = 11.

33. Types of Equations: Identity, Conditional, Inconsistent
An equation that is true for at least one real number is called a conditional equation. The examples that we have worked so far have been conditional equations.
An equation that is true for all real numbers is called an identity.
An equation that is not true for any real number is called an inconsistent equation.

34. Example: Categorizing an Equation
This is a false statement. This equation is
an example of an inconsistent equation.
There is no solution.

35. Example: Categorizing an Equation
This is a true statement. The solution set
for this equation is the set of all real numbers.
This equation is an identity.

36. Example: Solving Applied Problems Using Mathematical Models
Persons with a low sense of humor have higher levels of
depression in response to negative life events than those
with a high sense of humor. This can be modeled by the
formula In this formula, x represents
the intensity of a negative life event and D is the average level of depression in response to that event. If the low humor group averages a level of depression of 10 in response to a negative life event, what is the intensity of that event?

37. Example: Solving Applied Problems Using Mathematical Models (continued)
The intensity of the event is 3.7

38. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.3 Models and Applications
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 38

39. Objectives:
Use linear equations to solve problems.
Solve a formula for a variable.

40. Problem Solving with Linear Equations
A model is a mathematical representation of a real-world
situation. We will use the following general steps in
solving word problems:
Step 1 Let x represent one of the unknown quantities.
Step 2 Represent other unknown quantities in terms of x.
Step 3 Write an equation in x that models the conditions.
Step 4 Solve the equation and answer the question.
Step 5 Check the proposed solution in the original
wording of the problem.

41. Example: Using Linear Equations to Solve Problems
The median starting salary of a computer science major
exceeds that of an education major by $21 thousand. The
median starting salary of an economics major exceeds
that of an education major by $14 thousand. Combined,
their median starting salaries are $140 thousand. Determine
the median starting salaries of education majors, computer
science majors, and economics majors with bachelor’s
degrees.

42. Example: Using Linear Equations to Solve Problems (continued)
Step 1 Let x represent one of the unknown quantities.
x = median starting salary of an education major
Step 2 Represent other unknown quantities in terms of x.
x + 21 = median starting salary of a computer science major
x + 14 = median starting salary of an economics major
Step 3 Write an equation in x that models the conditions.

43. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
starting salary of an education major is x = 35
starting salary of a computer science major is x + 21 = 56
starting salary of an economics major is x + 14 = 49

44. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
(continued)
The median starting salary of an education major is
$35 thousand, the median starting salary of a computer
science major is $56 thousand, and the median starting
salary of an economics major is $49 thousand.

45. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem states that combined, the median starting
salaries are $140 thousand. Using the median salaries we
determined in Step 4, the sum is
$35 thousand + $56 thousand + $49 thousand,
or $140 thousand, which verifies the problem’s conditions.

46. Example: Using Linear Equations to Solve Problems
You are choosing between two texting plans. Plan A has a
monthly fee of $15 with a charge of $0.08 per text. Plan B
has a monthly fee of $3 with a charge of $0.12 per text. For
how many text messages will the costs for the two plans be
the same?
Step 1 Let x represent one of the unknown quantities.
x = charge per text
Step 2 Represent other unknown quantities in terms of x.
Plan A = 0.08x +15
Plan B = 0.12x + 3

47. Example: Using Linear Equations to Solve Problems (continued)
Step 3 Write an equation in x that models the conditions.
Step 4 Solve the equation and answer the question.
The cost of the two plans will be the same for
300 text messages.

48. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem asks for how many text messages will the
two plans be the same?
The cost of Plan A for 300 text messages is
0.08(300) + 15 = = 39
The cost of Plan B for 300 text messages is
0.12(300) + 3 = = 39
For 300 text messages, the plan costs are equal,
which verifies the problem’s conditions.

49. Example: Using Linear Equations to Solve Problems
After a 30% price reduction, you purchase a new computer
for $840. What was the computer’s price before the
reduction?
Step 1 Let x represent one of the unknown quantities.
x = price before reduction
Step 2 Represent other unknown quantities in terms of x.
price of new computer = x – 0.3x
Step 3 Write an equation in x that models the conditions.

50. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
The price of the computer before the reduction was $1200.
Step 5 Check the proposed solution in the original
wording of the problem.
The price before the reduction, $1200, minus the 30%
reduction should equal the reduced price given in the original
wording, $840.

51. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem. (continued)
This verifies that the computer’s price before the reduction
was $1200.

52. Example: Using Linear Equations to Solve Problems
You inherited $5000 with the stipulation that for the first year
the money had to be invested in two funds paying 9% and
11% annual interest. How much did you invest at each rate
if the total interest earned for the year was $487?
Step 1 Let x represent one of the unknown quantities.
x = the amount invested at 9%
Step 2 Represent other unknown quantities in terms of x.
5000 – x = the amount invested at 11%
Step 3 Write an equation in x that models the conditions.

53. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
the amount invested at 9% = x = 3150
the amount invested at 11% = 5000 – x = 1850
You should invest $3150 at 9% and $1850 at 11%.

54. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem states that the total interest from the dual
investments should be $487.
The interest earned at 9% is 0.09(3150) =
The interest earned at 11% is 0.11(1850) =
The total interest is = This
verifies that the amount invested at 9% should be
$3150 and that the amount invested at 11% should
be $1850.

55. Example: Using Linear Equations to Solve Problems
The length of a rectangular basketball court is 44 feet more
than the width. If the perimeter of the basketball court is
288 feet, what are its dimensions?
Step 1 Let x represent one of the unknown quantities.
x = the width of the basketball court
Step 2 Represent other unknown quantities in terms of x.
x + 44 = the length of the basketball court
Step 3 Write an equation in x that models the conditions.

56. Example: Using Linear Equations to Solve Problems (continued)
Step 4 Solve the equation and answer the question.
the width of the basketball court is x = 50 ft
the length of the basketball court is x + 44 = 94 ft
The dimensions of the basketball court are 50 ft by 94 ft

57. Example: Using Linear Equations to Solve Problems (continued)
Step 5 Check the proposed solution in the original
wording of the problem.
The problem states that the perimeter of the basketball court
is 288 feet. If the dimensions are 50 ft by 94 ft, then the
perimeter is 2(50) + 2(94) = = This
verifies the conditions of the problem.

58. Solving a Formula For A Variable
Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation.

59. Example: Solving a Formula for a Variable

60. Example: Solving a Formula for a Variable That Occurs Twice

61. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.4 Complex Numbers
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 61

62. Objectives:
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers.
Perform operations with square roots of negative
numbers.

63. Complex Numbers and Imaginary Numbers
The imaginary unit i is defined as
The set of all numbers in the form
a + bi
with real numbers a and b, and i, the imaginary unit,
is called the set of complex numbers.
The standard form of a complex number is
a + bi.

64. Operations on Complex Numbers
The form of a complex number a + bi is like the binomial
a + bx. To add, subtract, and multiply complex numbers,
we use the same methods that we use for binomials.

65. Example: Adding and Subtracting Complex Numbers
Perform the indicated operations, writing the result
in standard form:

66. Example: Multiplying Complex Numbers
Find the product:

67. Conjugate of a Complex Number
For the complex number a + bi, we define its complex
conjugate to be a – bi.
The product of a complex number and its conjugate
is a real number.

68. Complex Number Division
The goal of complex number division is to obtain a real
number in the denominator. We multiply the numerator
and denominator of a complex number quotient by the
conjugate of the denominator to obtain this real number.

69. Example: Using Complex Conjugates to Divide Complex Numbers
Divide and express the result in standard form:
In standard form, the result is

70. Principal Square Root of a Negative Number
For any positive real number b, the principal square root of
the negative number – b is defined by

71. Example: Operations Involving Square Roots of Negative Numbers
Perform the indicated operations and write the result in
standard form:

72. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.5 Quadratic Equations
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 72

73. Objectives:
Solve quadratic equations by factoring.
Solve quadratic equations by the square root property.
Solve quadratic equations by completing the square.
Solve quadratic equations using the quadratic formula.
Use the discriminant to determine the number and type of solutions.
Determine the most efficient method to use when solving a quadratic equation.
Solve problems modeled by quadratic equations.

74. Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written
in the general form
where a, b, and c are real numbers, with
A quadratic equation in x is also called a second-degree
polynomial equation in x.

75. The Zero-Product Principle
To solve a quadratic equation by factoring, we apply the zero-product principle which states that:
If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.
If AB = 0, then A = 0 or B = 0.

76. Solving a Quadratic Equation by Factoring
1. If necessary, rewrite the equation in the general form
, moving all nonzero terms to one side, thereby obtaining zero on the other side.
2. Factor completely.
3. Apply the zero-product principle, setting each factor containing a variable equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation.

77. Example: Solving Quadratic Equations by Factoring
Solve by factoring:
Step 1 Move all nonzero terms to one side and obtain zero on the other side.
Step 2 Factor

78. Example: Solving Quadratic Equations by Factoring (continued)
Steps 3 and 4 Set each factor equal to zero and solve the resulting equations.

79. Example: Solving Quadratic Equations by Factoring (continued)
Step 5 Check the solutions in the original equation.
Check

80. Solving Quadratic Equations by the Square Root Property
Quadratic equations of the form u2 = d, where u is an algebraic expression and d is a nonzero real number, can be solved by the Square Root Property:
If u is an algebraic expression and d is a nonzero real number, then u2 = d has exactly two solutions: or
Equivalently,
If u2 = d, then

81. Example: Solving Quadratic Equations by the Square Root Property
Solve by the square root property:

82. Completing the Square
If x2 + bx is a binomial, then by adding , which is
the square of half the coefficient of x, a perfect square trinomial will result. That is,

83. Example: Solving a Quadratic Equation by Completing the Square
Solve by completing the square:

84. The Quadratic Formula
The solutions of a quadratic equation in general form
with , are given by the quadratic formula:

85. Example: Solving a Quadratic Equation Using the Quadratic Formula
Solve using the quadratic formula:
a = 2, b = 2, c = – 1

86. Example: Solving a Quadratic Equation Using the Quadratic Formula (continued)

87. The Discriminant
We can find the solution for a quadratic equation of the form using the quadratic formula:
The discriminant is the quantity
which appears under the radical sign in the quadratic formula. The discriminant of the quadratic equation determines the number and type of solutions.

88. The Discriminant and the Kinds of Solutions to
If the discriminant is positive, there will be two unequal real solutions.
If the discriminant is zero, there is one real (repeated) solution.
If the discriminant is negative, there are two imaginary solutions.

89. Example: Using the Discriminant
Compute the discriminant, then determine the number and type of solutions:
The discriminant, 81, is a positive number. There are two real solutions.

90. Example: Application
The formula
models a woman’s normal systolic blood pressure, P, at age A. Use this formula to find the age, to the nearest year, of a woman whose normal systolic blood pressure is 115 mm Hg.
Solution: We will solve the equation

91. Example: Application (continued)

92. Example: Application (continued)
The positive solution,
indicates that 26 is the
approximate age of a woman
whose normal systolic blood
pressure is 115 mm Hg.

93. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.6 Other Types of
Equations
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 93

94. Objectives:
Solve polynomial equations by factoring.
Solve radical equations.
Solve equations with rational exponents.
Solve equations that are quadratic in form.
Solve equations involving absolute value.
Solve problems modeled by equations.

95. Polynomial Equations
A polynomial equation is the result of setting two polynomials equal to each other.
The equation is in general form if one side is 0 and the polynomial on the other side is in descending powers of the variable.
The degree of a polynomial equation is the same as the highest degree of any term in the equation.

96. Example: Solving a Polynomial Equation by Factoring
Solve by factoring:
Step 1 Move all nonzero terms to one side and obtain zero on the other side.
Step 2 Factor.

97. Example: Solving a Polynomial Equation by Factoring (continued)
Steps 3 and 4 Set each factor equal to zero and solve the resulting equations.
The solution set is
Step 5 Check the solutions in the original equation.

98. Radical Equations
A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root.
We solve radical equations with nth roots by raising both sides of the equation to the nth power.

99. Solving Radical Equations Containing nth Roots
1. If necessary, arrange terms so that one radical is isolated on one side of the equation.
2. Raise both sides of the equation to the nth power to eliminate the isolated nth root.
3. Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2.
4. Check all proposed solutions in the original equation.

100. Example: Solving a Radical Equation
Solve:
Step 1 Isolate a radical on one side.
Step 2 Raise both sides to the nth power.

101. Example: Solving a Radical Equation (continued)
Step 3 Solve the resulting equation

102. Example: Solving a Radical Equation (continued)
Step 4 Check the proposed solutions in the original equation.
Check 6: Check 1:
1 is an extraneous solution. The only solution is x = 6.

103. Equations with Rational Exponents
We know that rational exponents represent radicals:
A radical equation with rational exponents can be solved by isolating the expression with the rational exponent, and raising both sides of the equation to a power that is the reciprocal of the rational exponent.

104. Example: Solving Equations Involving Rational Exponents
Solve:

105. Equations That Are Quadratic in Form
An equation that is quadratic in form is one that can be expressed as a quadratic equation using an appropriate substitution.

106. Example: Solving an Equation Quadratic in Form
Solve:
Notice that , we let u = x2

107. Equations Involving Absolute Value
The absolute value of x describes the distance of x from zero on a number line. To solve an absolute value equation, we rewrite the absolute value equation without absolute value bars.
If c is a positive real number and u represents an algebraic expression, then is equivalent to u = c or u = – c.

108. Example: Solving an Equation Involving Absolute Value
Solve:
The solution set is

109. Example: Applications
The formula models weekly television viewing time, H, in hours, by annual income, I, in thousands of dollars. What annual income corresponds to 33.1 hours per week watching TV?
An annual income of $225,000 corresponds to 33.1 hours per week watching TV.

110. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1
Equations and
Inequalities
1.7 Linear Inequalities and
Absolute Value Inequalities
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
110

111. Objectives:
Use interval notation.
Find intersections and unions of intervals.
Solve linear inequalities.
Recognize inequalities with no solution or all real numbers as solutions.
Solve compound inequalities.
Solve absolute value inequalities.

112. Solving an Inequality
Solving an inequality is the process of finding the set of numbers that make the inequality a true statement. These numbers are called the solutions of the inequality and we say that they satisfy the inequality. The set of all solutions is called the solution set of the inequality.

113. Interval Notation
The open interval (a,b) represents the set of real numbers between, but not including, a and b.
The closed interval [a,b] represents the set of real numbers between, and including, a and b.

114. Interval Notation (continued)
The infinite interval represents the set of real numbers that are greater than a.
The infinite interval represents the set of real numbers that are less than or equal to b.

115. Parentheses and Brackets in Interval Notation
Parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpoints that are included in an interval. Parentheses are always used with or

116. Example: Using Interval Notation
Express the interval in set-builder notation and graph:
[1, 3.5]

117. Finding Intersections and Unions of Two Intervals
1. Graph each interval on a number line.
2. a. To find the intersection, take the portion of the
number line that the two graphs have in common.
b. To find the union, take the portion of the number
line representing the total collection of numbers
in the two graphs.

118. Example: Finding Intersections and Unions of Intervals
Use graphs to find the set:
Graph of [1,3]:
Graph of (2,6):
Numbers in both [1,3] and (2,6):
Thus,

119. Solving Linear Inequalities in One Variable
A linear inequality in x can be written in one of the following forms :
In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed.

120. Example: Solving a Linear Inequality
Solve and graph the solution set on a number line:
The solution set is
The interval notation for this
solution set is
The number line graph is:

121. Example: Solving Linear Inequalities [Recognize inequalities with no solution or all real numbers as solutions]
Solve the inequality:
The inequality is true for all values of x. The solution set
is the set of all real numbers.
In interval notation, the solution is
In set-builder notation, the solution set is

122. Example: Solving a Compound Inequality
Solve and graph the solution set on a number line:
Our goal is to isolate x in the middle.
In interval notation,
the solution is [-1,4).
In set-builder notation, the solution set is
The number line graph looks like

123. Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive number,
1. The solutions of are the numbers that satisfy
2. The solutions of are the numbers that satisfy or
These rules are valid if is replaced by and
is replaced by

124. Example: Solving an Absolute Value Inequality
Solve and graph the solution set on a number line:
We begin by expressing the inequality with the
absolute value expression on the left side:
We rewrite the inequality without absolute value bars.
means or

125. Example: Solving an Absolute Value Inequality (continued)
We solve these inequalities separately:
The solution set is
The number line graph looks like