Download 1.4 Linear Inequalities in One Variable

Section 1.4
Linear Inequalities in One Variable 81
1.4 Linear Inequalities in One Variable
OBJECTIVES
1
Represent Inequalities Using the
Real Number Line and Interval
Notation
2
Understand the Properties of
Inequalities
3
Solve Linear Inequalities
4
Solve Problems Involving Linear
Inequalities
Preparing for Linear Inequalities in One Variable
Before getting started, take the following readiness quiz. If you get a problem wrong, go back to the
section cited and review the material.
In Problems P1–P4, replace the question mark by 6 , 7 , or = to make the statement true.
[Section R.2, pp. 16–17]
2 3
1
P4. ?
? 0.5
2
3 5
P5. True or False: The inequality Ú is called a strict inequality. [Section R.2, p. 17]
P6. Use set-builder notation to represent the set
of all digits that are divisible by 3.
[Section R.2, pp. 8–9]
P1. 3 ? 6
P2. -3 ? -6
P3.
An inequality in one variable is a statement involving two expressions, at least one containing the variable, separated by one of the inequality symbols 6, …, 7, or Ú. To
solve an inequality means to find all values of the variable for which the statement is
true. These values are called solutions of the inequality. The set of all solutions is called
the solution set.
DEFINITION
A linear inequality in one variable is an inequality that can be written in the form
ax + b 6 c or ax + b … c or ax + b 7 c or ax + b Ú c
where a, b, and c are real numbers and a Z 0.
For example, the following are all linear inequalities involving one variable:
x - 4 7 9
5x - 1 … 14
8z 6 0
5x - 1 Ú 3x + 8
Before we discuss methods for solving linear inequalities, we will present three ways of
representing the solution set. One of the methods for representing the solution set is
through set-builder notation—something we are already familiar with. However, setbuilder notation can be somewhat cumbersome, so we introduce a more streamlined
way to represent a solution set to an inequality using interval notation. Finally, because
we often like to visualize solution sets, we present a method for graphing the solution
set on a real number line.
1
Represent Inequalities Using the Real Number Line
and Interval Notation
Suppose that a and b are two real numbers and a 6 b. We shall use the notation
a 6 x 6 b
Work Smart
Remember that the inequalities 6
and 7 are called strict inequalities,
while … and Ú are called nonstrict
inequalities.
Preparing for...Answers P1. 6 P2. 7
P3. = P4. 7 P5. False P6. 5x | x is
a digit that is divisible by 36
to mean that x is a number between a and b. So, the expression a 6 x 6 b is equivalent
to the two inequalities a 6 x and x 6 b. Similarly, the expression a … x … b is equivalent to the two inequalities a … x and x … b. We define a … x 6 b and a 6 x … b
similarly. Expressions such as -2 6 x 6 5 or x Ú 5 are said to be in inequality
notation.
While the expression 3 Ú x Ú 2 is technically correct, it is not the preferred way to
write an inequality. For ease of reading, we prefer that the numbers in the inequality go
from smaller values to larger values. So, we would write 3 Ú x Ú 2 as 2 … x … 3.
A statement such as 3 … x … 1 is false because there is no number x for which
3 … x and x … 1. We also never mix inequalities as in 2 … x Ú 3.
In addition to representing inequalities using inequality notation, we can use
interval notation.
82
CHAPTER 1 Linear Equations and Inequalities
DEFINITION INTERVAL NOTATION
Let a and b represent two real numbers with a 6 b.
A closed interval, denoted by [a, b], consists of all real numbers x for which
a … x … b.
An open interval, denoted by (a, b), consists of all real numbers x for which
a 6 x 6 b.
The half-open, or half-closed, intervals are (a, b], consisting of all real numbers x for
which a 6 x … b, and [a, b), consisting of all real numbers x for which a … x 6 b.
In each of these definitions, a is called the left endpoint and b is called the right
endpoint of the interval.
The symbol q (read as “infinity”) is not a real number, but a notational device used
to indicate unboundedness in the positive direction. In other words, the symbol q
means that there is no right endpoint on the inequality. The symbol - q (read as
“minus infinity” or “negative infinity”) also is not a real number, but a notational device used to indicate unboundedness in the negative direction. The symbol - q means
that there is no left endpoint on the inequality. Using the symbols q and - q , we can
define five other kinds of intervals.
Work Smart
The symbols q and - q are never
included as endpoints because they
are not real numbers. So, we use
parentheses when - q or q are
endpoints.
Figure 9
x⬎3
⫺1
0
1
2
3
4
5
1
2
3
4
5
Figure 10
x Ú 3
⫺1
0
INTERVALS INCLUDING ˆ
[a, q 2
1a, q 2
1 - q , a]
1 - q , a2
1- q, q2
consists of all real numbers x for which x Ú a
consists of all real numbers x for which x 7 a
consists of all real numbers x for which x … a
consists of all real numbers x for which x 6 a
consists of all real numbers x (or - q 6 x 6 q )
In addition to representing inequalities using interval notation, we can represent inequalities using a graph on the real number line. The inequality x 7 3 or the interval
(3, q ) consists of all numbers x that lie to the right of 3 on the real number line. We can
represent these values by shading the real number line to the right of 3.To indicate that 3
is not included in the set, we will agree to use a parenthesis on the endpoint. See Figure 9.
To represent the inequality x Ú 3 or the interval [3, q 2 graphically, we also shade
to the right of 3, but this time we use a bracket on the endpoint to indicate that 3 is
included in the set. See Figure 10.
Table 7 summarizes interval notation, inequality notation, and their graphs.
Table 7
Interval Notation
Inequality Notation
The open interval (a, b)
5x | a 6 x 6 b6
The closed interval [a, b]
5x | a … x … b6
The half-open interval [a, b)
5x | a … x 6 b6
The half-open interval (a, b]
5x | a 6 x … b6
The interval [a, q )
5x | x Ú a6
The interval (a, q )
5x | x 7 a6
The interval (- q , a]
5x | x … a6
The interval ( - q , a)
5x | x 6 a6
The interval (- q , q )
5x | x is a real number6
Graph
a
b
a
b
a
b
a
b
a
a
a
a
Section 1.4
Linear Inequalities in One Variable 83
EXAMPLE 1 Using Interval Notation and Graphing Inequalities
Write each inequality using interval notation. Graph the inequality.
(a) -2 … x … 4
(b) 1 6 x … 5
Solution
(a) -2 … x … 4 describes all numbers x between -2 and 4, inclusive. In interval notation, we write [-2, 4]. To graph -2 … x … 4, we place brackets at -2 and 4 and shade in between. See Figure 11.
Figure 11
⫺3 ⫺2 ⫺1
0
1
2
3
4
5
(b) 1 6 x … 5 describes all numbers x greater than 1 and less than or
equal to 5. In interval notation, we write (1, 5]. To graph 1 6 x … 5, we
place a parenthesis at 1 and a bracket at 5 and shade in between. See
Figure 12.
Figure 12
⫺1
0
1
2
3
4
5
6
EXAMPLE 2 Using Interval Notation and Graphing Inequalities
Write each inequality using interval notation. Graph the inequality.
(a) x 6 2
(b) x Ú -3
Solution
(a) x 6 2 describes all numbers x less than 2. In interval notation, we write
1- q , 22. To graph x 6 2, we place a parenthesis at 2 and then shade to
the left. See Figure 13.
Figure 13
⫺2 ⫺1
0
1
2
3
4
(b) x Ú -3 describes all numbers x greater than or equal to -3. In interval
notation, we write [-3, q 2. To graph x Ú -3, we place a bracket at -3
and then shade to the right. See Figure 14.
Figure 14
⫺4 ⫺3 ⫺2 ⫺1
0
1
2
3
Quick
1. A(n)
, denoted [a, b], consists of all real numbers x for which
a … x … b.
2. In the interval (a, b), a is called the
and b is called the
of the interval.
In Problems 3–6, write each inequality in interval notation. Graph the inequality.
3. -3 … x … 2
4. 3 … x 6 6
5. x … 3
6.
1
7
6 x 6
2
2
84 CHAPTER 1
Linear Equations and Inequalities
EXAMPLE 3 Using Inequality Notation and Graphing Inequalities
Write each interval in inequality notation involving x. Graph the inequality.
(a) [-2, 42
(b) (1, 5)
Solution
(a) The interval [-2, 42 consists of all numbers x for which -2 … x 6 4. See
Figure 15 for the graph.
(b) The interval (1, 5) consists of all numbers x for which 1 6 x 6 5. See
Figure 16 for the graph.
Figure 15
Figure 16
⫺3 ⫺2 ⫺1
0
1
2
3
4
⫺1
5
0
1
2
3
4
5
6
7
EXAMPLE 4 Using Inequality Notation and Graphing Inequalities
Write each interval in inequality notation involving x. Graph the inequality.
(a) c
3
, qb
2
(b) 1- q , 12
Solution
(a) The interval c
3
3
, q b consists of all numbers x for which x Ú .
2
2
See Figure 17 for the graph.
(b) The interval 1- q , 12 consists of all numbers x for which x 6 1. See
Figure 18 for the graph.
Figure 17
⫺1
0
Figure 18
1 _3_ 2
3
4
5
6
⫺4 ⫺3 ⫺2 ⫺1
7
0
1
2
3
4
2
Quick
In Problems 7–10, write each interval as an inequality. Graph the
inequality.
7. (0, 5]
8. 1-6, 02
9. 15, q 2
10. a - q ,
2
8
d
3
Understand the Properties of Inequalities
Consider the inequality 2 6 5. If we add 3 to both sides of the inequality, the expression
on the left becomes 5 and the expression on the right becomes 8. Since 5 6 8, we can see
that adding the same quantity to both sides of an inequality does not change the sense,
or direction, of the inequality. This result is called the Addition Property of Inequalities.
In Words
The Addition Property states that the
direction of the inequality does not change
when the same quantity is added to each side
of the inequality.
ADDITION PROPERTY OF INEQUALITIES
For real numbers a, b, and c,
If a 6 b,
If a 7 b,
then
then
a + c 6 b + c
a + c 7 b + c
For example, since 2 6 5, we have that 2 + 4 6 5 + 4 or 6 6 9. In addition, since
3 7 -1, we have that 3 + 1-22 7 -1 + 1-22 or 1 7 -3.
Because a - b is equivalent to a + (-b), the Addition Property of Inequalities can
also be used to subtract a real number from each side of an inequality without changing the direction of the inequality.
Section 1.4
Linear Inequalities in One Variable 85
We’ve seen what happens when we add a real number to both sides of an inequality.
What happens when we multiply both sides by a nonzero constant? Let’s see.
Consider the inequality 3 6 5. Multiply both sides of the inequality by 2. The expression on the left side of the inequality becomes 2132 = 6 and the expression on the
right becomes 2152 = 10. Certainly 6 6 10, so the direction of the inequality did not
change.
Again consider the inequality 3 6 5. Now multiply both sides of the inequality by
-2. The expression on the left side of the inequality becomes -2132 = -6 and the
expression on the right becomes -2152 = -10. Because -6 7 -10 we see that the
direction of the inequality is reversed.
These results are true in general and lead us to the Multiplication Properties of
Inequalities.
In Words
The Multiplication Property states that if both
sides of an inequality are multiplied by a
positive real number, the direction of the
inequality is unchanged. If both sides of
an inequality are multiplied by a negative
real number, the direction of the inequality is
reversed.
MULTIPLICATION PROPERTIES OF INEQUALITIES
Let a, b, and c be real numbers.
If a 6 b and if c 7 0, then ac 6 bc
If a 7 b and if c 7 0, then ac 7 bc
If a 6 b and if c 6 0, then ac 7 bc
If a 7 b and if c 6 0, then ac 6 bc
a
1
Because the quotient is equivalent to a # , the Multiplication Properties of Inequalities
b
b
can also be used to divide both sides of an inequality by a nonzero real number.
Quick
11. Write the inequality that results by adding 5 to each side of the inequality 4 6 7.
What property of inequalities does this illustrate?
12. Write the inequality that results by subtracting 3 from each side of the inequality x + 3 7 -6. What property of inequalities does this illustrate?
13. Write the inequality that results by multiplying both sides of the inequality
1
2 6 8 by . What property of inequalities does this illustrate?
2
14. Write the inequality that results by dividing both sides of the inequality
-6 6 9 by -3. What property of inequalities does this illustrate?
15. Write the inequality that results by dividing both sides of the inequality
5x 6 30 by 5. What property of inequalities does this illustrate?
3
Solve Linear Inequalities
Two inequalities that have exactly the same solution set are called equivalent
inequalities. As with equations, one method for solving a linear inequality is to replace
it by a series of equivalent inequalities until an inequality with an obvious solution,
such as x 7 2, is obtained. We obtain equivalent inequalities by applying some of the
same operations as those used to find equivalent equations. The Addition Property and
Multiplication Properties form the basis for the procedures.
Although not essential, it is easier to read an inequality if the variable is placed on
the left side and the constant on the right. If the variable does end up on the right side
of the inequality, we can rewrite it with the variable on the left side using the fact that
In Words
If the sides of an inequality are interchanged,
the direction of the inequality reverses.
a 6 x
a 7 x
is equivalent to
and
is equivalent to
x 7 a
x 6 a
86 CHAPTER 1
Linear Equations and Inequalities
EXAMPLE 5 How to Solve a Linear Inequality
Solve the inequality 3x - 2 7 13. Graph the solution set.
Step-by-Step Solution
The goal in solving any linear inequality is to get the variable by itself with a coefficient of 1.
Step 1: Isolate the term containing the
Add 2 to both sides
(Addition Property):
variable.
Divide both sides by 3
(Multiplication Property):
Step 2: Get a coefficient of 1 on the
variable.
0
3x
15
7
3
3
x 7 5
The solution using set-builder notation is 5x | x 7 56. The solution using interval notation is 15, q 2. Figure 19 shows the graph of the solution set.
Figure 19
⫺1
3x - 2 7 13
3x - 2 + 2 7 13 + 2
3x 7 15
1
2
3
4
5
6
7
Quick
In Problems 16–19, solve each linear inequality. Express your solution
using set-builder notation and interval notation. Graph the solution set.
1
x … 2
3
19. -2x + 1 … 13
16. x + 3 7 5
17.
18. 4x - 3 6 13
EXAMPLE 6 Solving Linear Inequalities
Solve the inequality: x - 4 Ú 5x + 12
Solution
Add 4 to both sides:
Subtract 5x from both sides:
Divide both sides by ⫺4. Don’t forget to
change the direction of the inequality:
Ú
Ú
Ú
Ú
Ú
5x +
5x +
5x +
5x +
16
16
…
-4
… -4
12
12 + 4
16
16 - 5x
The solution using set-builder notation is 5x | x … -46. The solution using interval
notation is 1- q , -4]. See Figure 20 for the graph of the solution set.
Figure 20
⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
x - 4
x - 4 + 4
x
x - 5x
-4x
-4x
-4
x
0
1
Quick
In Problems 20–22, solve each linear inequality. Express your solution
using set-builder notation and interval notation. Graph the solution set.
20. 3x + 1 7 x - 5
22. -5x + 12 6 x - 3
21. -2x + 1 … 3x + 11
Section 1.4
Linear Inequalities in One Variable 87
EXAMPLE 7 Solving Linear Inequalities
Solve the inequality: 31x - 12 + 2x 6 6x + 3
Solution
31x - 12 + 2x 6 6x + 3
3x - 3 + 2x
5x - 3
5x - 3 + 3
5x
Distribute the 3:
Combine like terms:
Add 3 to both sides:
6
6
6
6
6x
6x
6x
6x
+
+
+
+
3
3
3 + 3
6
5x - 6x 6 6x + 6 - 6x
-x 6 6
Multiply both sides of the inequality by -1:
1-121-x2 7 1-126
Subtract 6x from both sides:
x 7 -6
The solution using set-builder notation is 5x | x 7 -66. The solution using interval
notation is 1-6, q 2. See Figure 21 for the graph of the solution set.
Figure 21
⫺7 ⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
1
Quick
In Problems 23–25, solve each linear inequality. Express your solution
using set-builder notation and interval notation. Graph the solution set.
23. 41x - 22 6 3x - 4
24. -21x + 12 Ú 41x + 32
25. 7 - 21x + 12 … 31x - 52
EXAMPLE 8 Solving Linear Inequalities Involving Fractions
Solve the inequality:
x - 2
2x + 1
7
3
2
Solution
We begin by rewriting the linear inequality without fractions by multiplying both
sides of the inequality by 6, the Least Common Denominator.
6# a
2x + 1
x - 2
b 7 6# a
b
3
2
212x + 12 7 31x - 22
Distribute:
Subtract 2 from both sides:
4x + 2 7 3x - 6
4x + 2 - 2 7 3x - 6 - 2
4x 7 3x - 8
4x - 3x 7 3x - 8 - 3x
x 7 -8
Subtract 3x from both sides:
The solution using set-builder notation is 5x | x 7 -86. The solution using interval
notation is 1-8, q 2. See Figure 22 for the graph of the solution set.
Figure 22
⫺10 ⫺8 ⫺6 ⫺4 ⫺2
0
2
4
6
8
88 CHAPTER 1
Linear Equations and Inequalities
Quick
In Problems 26–28, solve each linear inequality. Express your solution
using set-builder notation and interval notation. Graph the solution set.
26.
3x + 1
Ú 2
5
28.
1
1
1x + 32 7 1x - 42
2
3
4
Solve Problems Involving Linear Inequalities
27.
2
3
1
x +
6
5
10
2
When you are confronted with a word problem, one of the first things that you need to
do is look for key words that tip you off as to the type of word problem that it is. There
are certain phrases that frequently occur in problems that lead to linear inequalities.
We list some of these phrases for you in Table 8.
Table 8
Phrase
Inequality
Phrase
Inequality
At least
Ú
No more than
…
No less than
Ú
At most
…
More than
7
Fewer than
6
Greater than
7
Less than
6
When solving applications involving linear inequalities, we use the same steps for
setting up applied problems that we introduced in Section 1.2 on page 61.
EXAMPLE 9 Comparing Credit Cards
Chase Bank has offered you two different credit card options. The Southwest rewards
card charges an annual fee of $39 plus 12.90% simple interest on all outstanding balances. The Marriott rewards card charges an annual fee of $30 plus 14.15% simple interest on all outstanding balances. What annual balance results in the Southwest card
costing less than the Marriott card? (SOURCE: chase.com)
Solution
Step 1: Identify We want to know the credit card balance for which the Southwest
credit card costs less than the Marriott rewards card. The phrase “costs less” implies
that this is an inequality problem.
Step 2: Name Let b represent the credit card balance on each card.
Step 3: Translate Each card charges an annual fee plus simple interest. So, for each
card the cost will be “annual fee ⫹ interest.”
Recall from Section 1.2 that simple interest is found using the formula
I = Prt
where I is the interest charged, P is the balance on the credit card, r is the annual
interest rate, and t is time.
In this problem, we let b represent the credit card balance. The annual interest
rate r will either be 0.129 (for Southwest) or 0.1415 (for Marriott). Because we are
discussing annual cost, we have that t ⫽ 1.
Section 1.4
Linear Inequalities in One Variable 89
Since we want to know what balance results in Southwest costing less than Marriott,
we have the following inequality:
Southwest Credit Card Cost 6 Marriott Credit Card Cost
Annual Fee Interest Charged
for Southwest for Southwest
#
⫹
39
4
0.129b
Annual Fee
for Marriott
⬍
#
30
Interest Charged
for Marriott
⫹
2
0.1415b
The Model
Step 4: Solve Solve the inequality for b.
Subtract 39 from both sides:
Subtract 0.1415b from both sides:
Divide both sides by ⫺0.0125.
Don’t forget to reverse the
inequality symbol:
39 + 0.129b 6 30 + 0.1415b
0.129b 6 -9 + 0.1415b
-0.0125b 6 -9
b 7 720
Step 5: Check If the balance is $720, then the annual cost for Southwest is 39 ⫹
0.129(720) ⫽ $131.88. The annual cost for Marriott is 30 ⫹ 0.1415(720) ⫽ $131.88.
For a balance greater than $720, say $750, the annual cost for Southwest is 39 ⫹
0.129(750) ⫽ $135.75. The annual cost for Marriott is 30 ⫹ 0.1415(750) ⫽ $136.13.
Step 6: Answer the Question If the annual balance is greater than $720, then the
Southwest card offers a better deal than the Marriott card.
Quick
29. You have just received two credit card applications in the mail. The card from
Bank A has an annual fee of $25 and charges 9.9% simple interest. The card
from Bank B has no annual fee, but charges 14.9% simple interest. For what
annual balance will the card from Bank A cost less than the card from Bank B?
30. Suppose the daily revenue from selling x boxes of candy is given by the equation R = 12x. The daily cost of operating the store and making the candy is
given by the equation C = 8x + 96. For how many boxes of candy will revenue
exceed costs? That is, solve R 7 C.
1.4 EXERCISES
1–30. are the Quick
s that follow each EXAMPLE
Building Skills
In Problems 31–38, write each inequality using interval notation.
Graph the inequality. See Objective 1.
31. 2 … x … 10
32. 1 6 x 6 7
33. -4 … x 6 0
34. -8 6 x … 1
35. x Ú 6
36. x 6 0
37. x 6
3
2
38. x Ú -
5
2
In Problems 39–46, write each interval as an inequality involving x.
Graph each inequality. See Objective 1.
39. (1, 8)
40. [-2, 3]
41. (-5, 1]
42. [1, 4)
43. (- q , 5)
44. (2, q )
45. [3, q )
46. (- q , 8]