# Section Linear Inequalities and Absolute Objectives Value ... · PDF file Section P.9 Linear Inequalities and Absolute Value Inequalities 117 Finding Intersections and Unions of Two

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• 54. I find the hardest part in solving a word problem is writing the equation that models the verbal conditions.

55. After a 35% reduction, a computer’s price is \$780, so I determined the original price, by solving

56. When I use the square root property to determine the length of a right triangle’s side, I don’t even bother to list the negative square root.

57. The perimeter of a plot of land in the shape of a right triangle is 12 miles. If one leg of the triangle exceeds the other leg by 1 mile, find the length of each boundary of the land.

58. The price of a dress is reduced by 40%. When the dress still does not sell, it is reduced by 40% of the reduced price. If the price of the dress after both reductions is \$72, what was the original price?

59. In a film, the actor Charles Coburn plays an elderly “uncle” character criticized for marrying a woman when he is 3 times her age. He wittily replies, “Ah, but in 20 years time I shall only be twice her age.” How old is the “uncle” and the woman?

60. Suppose that we agree to pay you 8¢ for every problem in this chapter that you solve correctly and fine you 5¢ for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?

61. It was wartime when the Ricardos found out Mrs. Ricardo was pregnant. Ricky Ricardo was drafted and made out a will, deciding that \$14,000 in a savings account was to be divided between his wife and his child-to-be. Rather strangely, and certainly with gender bias, Ricky stipulated that if the child were a boy, he would get twice the amount of the mother’s portion. If it were a girl, the mother would get twice the amount the girl was to receive. We’ll never know what Ricky was thinking of, for (as fate would have it) he did not return from war. Mrs. Ricardo gave birth to twins—a boy and a girl. How was the money divided?

x - 0.35 = 780.x,

114 Chapter P Prerequisites: Fundamental Concepts of Algebra

62. A thief steals a number of rare plants from a nursery. On the way out, the thief meets three security guards, one after another. To each security guard, the thief is forced to give one-half the plants that he still has, plus 2 more. Finally, the thief leaves the nursery with 1 lone palm. How many plants were originally stolen?

Group Exercise 63. One of the best ways to learn how to solve a word problem in

algebra is to design word problems of your own. Creating a word problem makes you very aware of precisely how much information is needed to solve the problem. You must also focus on the best way to present information to a reader and on how much information to give. As you write your problem, you gain skills that will help you solve problems created by others.

The group should design five different word problems that can be solved using equations. All of the problems should be on different topics. For example, the group should not have more than one problem on the perimeter of a rectangle. The group should turn in both the problems and their algebraic solutions.

(If you’re not sure where to begin, consider the graph for Exercises 5–6 and the data that we did not use regarding attitudes about women in combat.)

Preview Exercises Exercises 64–66 will help you prepare for the material covered in the next section.

64. Is a solution of

65. Solve:

66. Solve: x + 3

4 =

x - 2 3

+ 1 4

.

-2x - 4 = x + 5.

3 - 2x … 11?-1

P.9 Linear Inequalities and Absolute Value Inequalities

Rent-a-Heap, a car rental com-pany, charges \$125 per week plus \$0.20 per mile to rent one of their cars. Suppose you are limited by how much money you can spend for the week: You can spend at most \$335. If we let represent the number of miles you drive the heap in a week, we can write an inequality that models the given conditions:

x

Objectives

� Use interval notation. � Find intersections and unions

of intervals.

� Solve linear inequalities. � Solve compound inequalities. � Solve absolute value

inequalities.

Sec t i on

the charge of \$0.20 per mile

for x miles

must be less than

or equal to The weekly

charge of \$125

125 + 0.20x � 335.

plus \$335.

P-BLTZMC0P_001-134-hr 14-11-2008 18:05 Page 114

• Section P.9 Linear Inequalities and Absolute Value Inequalities 115

Interval Notation Graph

The open interval represents the set of real numbers between, but not including, and

x is greater than a (a < x) and

x is less than b (x < b).

(a, b)={x | a � x � b}

b.a 1a, b2

The parentheses in the graph and in interval notation indicate that a and b, the endpoints, are

excluded from the interval.

b(a, b)a x

The closed interval represents the set of real numbers between, and including, and

x is greater than or equal to a (a ≤ x) and

x is less than or equal to b (x ≤ b).

[a, b]={x | a � x � b}

b.a 3a, b4

The square brackets in the graph and in interval notation indicate that a and b, the endpoints, are

included in the interval.

b[a, b]a x

The infinite interval represents the set of real numbers that are greater than

The infinity symbol does not represent a real number. It indicates that the interval

extends indefinitely to the right.

(a, �)={x | x a}

a. 1a, q2

The parenthesis indicates that a is excluded from the interval.

(a, �)a x

The infinite interval represents the set of real numbers that are less than or equal to

The negative infinity symbol indicates that the interval

extends indefinitely to the left.

(–�, b]={x | x � b}

b. 1- q , b4

The square bracket indicates that b is included in the interval.

(−�, b] b x

Placing an inequality symbol between a polynomial of degee 1 and a constant results in a linear inequality in one variable. In this section, we will study how to solve linear inequalities such as . 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. Set-builder notation and a new notation, called interval notation, are used to represent these solution sets. We begin this section by looking at interval notation.

Interval Notation Some sets of real numbers can be represented using interval notation. Suppose that

and are two real numbers such that a 6 b.ba

125 + 0.20x … 335

� Use interval notation.

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 - q .q

P-BLTZMC0P_001-134-hr 14-11-2008 18:05 Page 115

• 116 Chapter P Prerequisites: Fundamental Concepts of Algebra

Table P.7 lists nine possible types of intervals used to describe subsets of real numbers.

� Find intersections and unions of intervals.

Table P.7 Intervals on the Real Number Line

Let and be real numbers such that a

• Section P.9 Linear Inequalities and Absolute Value Inequalities 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.

Finding Intersections and Unions of Intervals

Use graphs to find each set:

a. b.

Solution

a. the intersection of the intervals (1, 4) and [2, 8], consists of the numbers that are in both intervals.

To find take the portion of the number line that the two graphs have in common.

Thus,

b. the union of the intervals (1, 4) and [2, 8], consists of the numbers that are in either one interval or the other (or both).

To find take the portion of the number line representing the total collection of numbers in the two graphs.

Thus,

Check Point 2 Use graphs to find each set: a. b. 31, 34 ´ 12, 62.31, 34 ¨ 12, 62

11, 42 ´ 32, 84 = 11, 84.

The numbers in either one interval or the other (or both) are those that

are greater than 1 and less than or equal to 8: (1, 8].

Numbers in either (1, 4) or [2, 8] or both:

1 2 3 40 106 7 8 95 x

{x � 1 < x < 4 or 2 ≤ x ≤ 8}

11, 42 ´ 32, 84,

Graph of (1, 4):

Graph of [2, 8]:

1 2 3 40 106 7 8 95 x

1 2 3 40 106 7 8 95 x

{x � 1 < x < 4}

{x � 2 ≤ x ≤ 8}

11, 42 ´ 32, 84, 11, 42 ¨ 32, 84 = 32, 42.

The numbers common to both intervals are those that are greater than or

equal to 2 and less than 4: [2, 4).

Numbers in both (1, 4) and [2, 8]:

1 2 3 40 106 7 8 95 x

{x � 1 < x < 4 and 2 ≤ x ≤ 8}

11, 42 ¨ 32, 84,

Graph of (1, 4):

Graph of [2, 8]:

1 2 3 40 106 7 8 95 x

1 2 3 40 106 7 8 95 x

{x � 1 < x < 4}

{x � 2 ≤ x ≤ 8}

11, 42 ¨ 32, 84,

11, 42 ´ 32, 84.11, 42 ¨ 32, 84

EXAMPLE 2

P-BLTZMC0P_001-134-hr

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