Elementary Algebra

# 2.7Solve Linear Inequalities

Elementary Algebra2.7 Solve Linear Inequalities

## Learning Objectives

By the end of this section, you will be able to:

• Graph inequalities on the number line
• Solve inequalities using the Subtraction and Addition Properties of inequality
• Solve inequalities using the Division and Multiplication Properties of inequality
• Solve inequalities that require simplification
• Translate to an inequality and solve

## Be Prepared 2.7

Before you get started, take this readiness quiz.

1. Translate from algebra to English: $15>x15>x$.
If you missed this problem, review Example 1.12.
2. Solve: $n−9=−42.n−9=−42.$
If you missed this problem, review Example 2.3.
3. Solve: $−5p=−23.−5p=−23.$
If you missed this problem, review Example 2.13.
4. Solve: $3a−12=7a−20.3a−12=7a−20.$
If you missed this problem, review Example 2.34.

## Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality $x>3x>3$ true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality $x>3x>3$.

We show the solutions to the inequality $x>3x>3$ on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of $x>3x>3$ is shown in Figure 2.7. Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.

Figure 2.7 The inequality $x>3x>3$ is graphed on this number line.

The graph of the inequality $x≥3x≥3$ is very much like the graph of $x>3x>3$, but now we need to show that 3 is a solution, too. We do that by putting a bracket at $x=3x=3$, as shown in Figure 2.8.

Figure 2.8 The inequality $x≥3x≥3$ is graphed on this number line.

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

## Example 2.66

Graph on the number line:

$x≤1x≤1$ $x<5x<5$ $x>−1x>−1$

## Try It 2.131

Graph on the number line: $x≤−1x≤−1$ $x>2x>2$ $x<3x<3$

## Try It 2.132

Graph on the number line: $x>−2x>−2$ $x<−3x<−3$ $x≥−1x≥−1$

We can also represent inequalities using interval notation. As we saw above, the inequality $x>3x>3$ means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express $x>3x>3$ as $(3,∞).(3,∞).$ The symbol $∞∞$ is read as ‘infinity’. It is not an actual number. Figure 2.9 shows both the number line and the interval notation.

Figure 2.9 The inequality $x>3x>3$ is graphed on this number line and written in interval notation.

The inequality $x≤1x≤1$ means all numbers less than or equal to 1. There is no lower end to those numbers. We write $x≤1x≤1$ in interval notation as $(−∞,1](−∞,1]$. The symbol $−∞−∞$ is read as ‘negative infinity’. Figure 2.10 shows both the number line and interval notation.

Figure 2.10 The inequality $x≤1x≤1$ is graphed on this number line and written in interval notation.

## Inequalities, Number Lines, and Interval Notation

Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure 2.11.

Figure 2.11 The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals.

## Example 2.67

Graph on the number line and write in interval notation.

$x≥−3x≥−3$ $x<2.5x<2.5$ $x≤−35x≤−35$

## Try It 2.133

Graph on the number line and write in interval notation:

$x>2x>2$ $x≤−1.5x≤−1.5$ $x≥34x≥34$

## Try It 2.134

Graph on the number line and write in interval notation:

$x≤−4x≤−4$ $x≥0.5x≥0.5$ $x<−23x<−23$

## Solve Inequalities using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

## Properties of Equality

$Subtraction Property of EqualityAddition Property of EqualityFor any numbersa,b,andc,For any numbersa,b,andc,ifa=b,thena−c=b−c.ifa=b,thena+c=b+c.Subtraction Property of EqualityAddition Property of EqualityFor any numbersa,b,andc,For any numbersa,b,andc,ifa=b,thena−c=b−c.ifa=b,thena+c=b+c.$

Similar properties hold true for inequalities.

 For example, we know that −4 is less than 2. If we subtract 5 from both quantities, is theleft side still less than the right side? We get −9 on the left and −3 on the right. And we know −9 is less than −3. The inequality sign stayed the same.

Similarly we could show that the inequality also stays the same for addition.

## Properties of Inequality

$Subtraction Property of InequalityAddition Property of InequalityFor any numbersa,b,andc,For any numbersa,b,andc,ifabthena−c>b−c.ifabthena+c>b+c.Subtraction Property of InequalityAddition Property of InequalityFor any numbersa,b,andc,For any numbersa,b,andc,ifabthena−c>b−c.ifabthena+c>b+c.$

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality $x+5>9x+5>9$, the steps would look like this:

 $x+5>9x+5>9$ Subtract 5 from both sides to isolate $xx$. $x+5−5>9−5x+5−5>9−5$ Simplify. $x>4x>4$
Table 2.6

Any number greater than 4 is a solution to this inequality.

## Example 2.68

Solve the inequality $n−12≤58n−12≤58$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.135

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$p−34≥16p−34≥16$

## Try It 2.136

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$r−13≤712r−13≤712$

## Solve Inequalities using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

## Properties of Equality

$Division Property of EqualityMultiplication Property of EqualityFor any numbersa,b,c,andc≠0,For any real numbersa,b,c,ifa=b,thenac=bc.ifa=b,thenac=bc.Division Property of EqualityMultiplication Property of EqualityFor any numbersa,b,c,andc≠0,For any real numbersa,b,c,ifa=b,thenac=bc.ifa=b,thenac=bc.$

Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?

Consider some numerical examples.

 Divide both sides by 5. Multiply both sides by 5. Simplify. Fill in the inequality signs.
$The inequality signs stayed the same.The inequality signs stayed the same.$

Does the inequality stay the same when we divide or multiply by a negative number?

 Divide both sides by −5. Multiply both sides by −5. Simplify. Fill in the inequality signs.
$The inequality signs reversed their direction.The inequality signs reversed their direction.$

When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.

Here are the Division and Multiplication Properties of Inequality for easy reference.

## Division and Multiplication Properties of Inequality

$For any real numbersa,b,c ifa0,thenacbandc>0,thenac>bcandac>bc. ifabcandac>bc. ifa>bandc<0,thenac0,thenacbandc>0,thenac>bcandac>bc. ifabcandac>bc. ifa>bandc<0,thenac

When we divide or multiply an inequality by a:

• positive number, the inequality stays the same.
• negative number, the inequality reverses.

## Example 2.69

Solve the inequality $7y<​​427y<​​42$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.137

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$(8,∞)(8,∞)$

## Try It 2.138

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$12d≤​6012d≤​60$

## Example 2.70

Solve the inequality $−10a≥50−10a≥50$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.139

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$−8q<32−8q<32$

## Try It 2.140

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$−7r≤​−70−7r≤​−70$

## Solving Inequalities

Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.

$x>ahas the same meaning asaahas the same meaning asa

Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”

## Example 2.71

Solve the inequality $−20<45u−20<45u$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.141

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$24≤38m24≤38m$

## Try It 2.142

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$−24<43n−24<43n$

## Example 2.72

Solve the inequality $t−2≥8t−2≥8$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.143

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$k−12≤15k−12≤15$

## Try It 2.144

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$u−4≥−16u−4≥−16$

## Solve Inequalities That Require Simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.

## Example 2.73

Solve the inequality $4m≤9m+174m≤9m+17$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.145

Solve the inequality $3q ≥ 7q − 233q ≥ 7q − 23$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.146

Solve the inequality $6x<10x+196x<10x+19$, graph the solution on the number line, and write the solution in interval notation.

## Example 2.74

Solve the inequality $8p+3(p−12)>7p−288p+3(p−12)>7p−28$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.147

Solve the inequality $9y+2(y+6)>5y−249y+2(y+6)>5y−24$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.148

Solve the inequality $6u+8(u−1)>10u+326u+8(u−1)>10u+32$, graph the solution on the number line, and write the solution in interval notation.

Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.

## Example 2.75

Solve the inequality $8x−2(5−x)<4(x+9)+6x8x−2(5−x)<4(x+9)+6x$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.149

Solve the inequality $4b−3(3−b)>5(b−6)+2b4b−3(3−b)>5(b−6)+2b$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.150

Solve the inequality $9h−7(2−h)<8(h+11)+8h9h−7(2−h)<8(h+11)+8h$, graph the solution on the number line, and write the solution in interval notation.

## Example 2.76

Solve the inequality $13a−18a>524a​+3413a−18a>524a​+34$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.151

Solve the inequality $14x−112x>16x+7814x−112x>16x+78$, graph the solution on the number line, and write the solution in interval notation.

## Try It 2.152

Solve the inequality $25z−13z<115z​−3525z−13z<115z​−35$, graph the solution on the number line, and write the solution in interval notation.

## Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.

Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.

Table 2.7 shows some common phrases that indicate inequalities.

$>>$ $≥≥$ $<<$ $≤≤$
is greater than is greater than or equal to is less than is less than or equal to
is more than is at least is smaller than is at most
is larger than is no less than has fewer than is no more than
exceeds is the minimum is lower than is the maximum
Table 2.7

## Example 2.77

Translate and solve. Then write the solution in interval notation and graph on the number line.

Twelve times c is no more than 96.

## Try It 2.153

Translate and solve. Then write the solution in interval notation and graph on the number line.

Twenty times y is at most 100

## Try It 2.154

Translate and solve. Then write the solution in interval notation and graph on the number line.

Nine times z is no less than 135

## Example 2.78

Translate and solve. Then write the solution in interval notation and graph on the number line.

Thirty less than x is at least 45.

## Try It 2.155

Translate and solve. Then write the solution in interval notation and graph on the number line.

Nineteen less than p is no less than 47

## Try It 2.156

Translate and solve. Then write the solution in interval notation and graph on the number line.

Four more than a is at most 15.

## Section 2.7 Exercises

### Practice Makes Perfect

Graph Inequalities on the Number Line

In the following exercises, graph each inequality on the number line.

430.

$x≤2x≤2$
$x>−1x>−1$
$x<0x<0$

431.

$x>1x>1$
$x<−2x<−2$
$x≥−3x≥−3$

432.

$x≥−3x≥−3$
$x<4x<4$
$x≤−2x≤−2$

433.

$x≤0x≤0$
$x>−4x>−4$
$x≥−1x≥−1$

In the following exercises, graph each inequality on the number line and write in interval notation.

434.

$x<−2x<−2$
$x≥−3.5x≥−3.5$
$x≤23x≤23$

435.

$x>3x>3$
$x≤−0.5x≤−0.5$
$x≥13x≥13$

436.

$x≥−4x≥−4$
$x<2.5x<2.5$
$x>−32x>−32$

437.

$x≤5x≤5$
$x≥−1.5x≥−1.5$
$x<−73x<−73$

Solve Inequalities using the Subtraction and Addition Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

438.

$n − 11 < 33 n − 11 < 33$

439.

$m − 45 ≤ 62 m − 45 ≤ 62$

440.

$u + 25 > 21 u + 25 > 21$

441.

$v + 12 > 3 v + 12 > 3$

442.

$a + 3 4 ≥ 7 10 a + 3 4 ≥ 7 10$

443.

$b + 7 8 ≥ 1 6 b + 7 8 ≥ 1 6$

444.

$f − 13 20 < − 5 12 f − 13 20 < − 5 12$

445.

$g − 11 12 < − 5 18 g − 11 12 < − 5 18$

Solve Inequalities using the Division and Multiplication Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

446.

$8 x > 72 8 x > 72$

447.

$6 y < 48 6 y < 48$

448.

$7 r ≤ 56 7 r ≤ 56$

449.

$9 s ≥ 81 9 s ≥ 81$

450.

$−5 u ≥ 65 −5 u ≥ 65$

451.

$−8 v ≤ 96 −8 v ≤ 96$

452.

$−9 c < 126 −9 c < 126$

453.

$−7 d > 105 −7 d > 105$

454.

$20 > 2 5 h 20 > 2 5 h$

455.

$40 < 5 8 k 40 < 5 8 k$

456.

$7 6 j ≥ 42 7 6 j ≥ 42$

457.

$9 4 g ≤ 36 9 4 g ≤ 36$

458.

$a −3 ≤ 9 a −3 ≤ 9$

459.

$b −10 ≥ 30 b −10 ≥ 30$

460.

$−25 < p −5 −25 < p −5$

461.

$−18 > q −6 −18 > q −6$

462.

$9 t ≥ −27 9 t ≥ −27$

463.

$7 s < − 28 7 s < − 28$

464.

$2 3 y > − 36 2 3 y > − 36$

465.

$3 5 x ≤ − 45 3 5 x ≤ − 45$

Solve Inequalities That Require Simplification

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

466.

$4 v ≥ 9 v − 40 4 v ≥ 9 v − 40$

467.

$5 u ≤ 8 u − 21 5 u ≤ 8 u − 21$

468.

$13 q < 7 q − 29 13 q < 7 q − 29$

469.

$9 p > 14 p − 18 9 p > 14 p − 18$

470.

$12 x + 3 ( x + 7 ) > 10 x − 24 12 x + 3 ( x + 7 ) > 10 x − 24$

471.

$9 y + 5 ( y + 3 ) < 4 y − 35 9 y + 5 ( y + 3 ) < 4 y − 35$

472.

$6 h − 4 ( h − 1 ) ≤ 7 h − 11 6 h − 4 ( h − 1 ) ≤ 7 h − 11$

473.

$4 k − ( k − 2 ) ≥ 7 k − 26 4 k − ( k − 2 ) ≥ 7 k − 26$

474.

$8 m − 2 ( 14 − m ) ≥ ​ 7 ( m − 4 ) + 3 m 8 m − 2 ( 14 − m ) ≥ ​ 7 ( m − 4 ) + 3 m$

475.

$6 n − 12 ( 3 − n ) ≤ 9 ( n − 4 ) + 9 n 6 n − 12 ( 3 − n ) ≤ 9 ( n − 4 ) + 9 n$

476.

$3 4 b − 1 3 b < 5 12 b − 1 2 3 4 b − 1 3 b < 5 12 b − 1 2$

477.

$9 u + 5 ( 2 u − 5 ) ≥ 12 ( u − 1 ) + 7 u 9 u + 5 ( 2 u − 5 ) ≥ 12 ( u − 1 ) + 7 u$

478.

$2 3 g − 1 2 ( g − 14 ) ≤ 1 6 ( g + 42 ) 2 3 g − 1 2 ( g − 14 ) ≤ 1 6 ( g + 42 )$

479.

$5 6 a − 1 4 a > 7 12 a + 2 3 5 6 a − 1 4 a > 7 12 a + 2 3$

480.

$4 5 h − 2 3 ( h − 9 ) ≥ 1 15 ( 2 h + 90 ) 4 5 h − 2 3 ( h − 9 ) ≥ 1 15 ( 2 h + 90 )$

481.

$12 v + 3 ( 4 v − 1 ) ≤ 19 ( v − 2 ) + 5 v 12 v + 3 ( 4 v − 1 ) ≤ 19 ( v − 2 ) + 5 v$

Mixed practice

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

482.

$15 k ≤ − 40 15 k ≤ − 40$

483.

$35 k ≥ −77 35 k ≥ −77$

484.

$23 p − 2 ( 6 − 5 p ) > 3 ( 11 p − 4 ) 23 p − 2 ( 6 − 5 p ) > 3 ( 11 p − 4 )$

485.

$18 q − 4 ( 10 − 3 q ) < 5 ( 6 q − 8 ) 18 q − 4 ( 10 − 3 q ) < 5 ( 6 q − 8 )$

486.

$− 9 4 x ≥ − 5 12 − 9 4 x ≥ − 5 12$

487.

$− 21 8 y ≤ − 15 28 − 21 8 y ≤ − 15 28$

488.

$c + 34 < − 99 c + 34 < − 99$

489.

$d + 29 > − 61 d + 29 > − 61$

490.

$m 18 ≥ −4 m 18 ≥ −4$

491.

$n 13 ≤ − 6 n 13 ≤ − 6$

Translate to an Inequality and Solve

In the following exercises, translate and solve .Then write the solution in interval notation and graph on the number line.

492.

Fourteen times d is greater than 56.

493.

Ninety times c is less than 450.

494.

Eight times z is smaller than $−40−40$.

495.

Ten times y is at most $−110−110$.

496.

Three more than h is no less than 25.

497.

Six more than k exceeds 25.

498.

Ten less than w is at least 39.

499.

Twelve less than x is no less than 21.

500.

Negative five times r is no more than 95.

501.

Negative two times s is lower than 56.

502.

Nineteen less than b is at most $−22−22$.

503.

Fifteen less than a is at least $−7−7$.

### Everyday Math

504.

Safety A child’s height, h, must be at least 57 inches for the child to safely ride in the front seat of a car. Write this as an inequality.

505.

Fighter pilots The maximum height, h, of a fighter pilot is 77 inches. Write this as an inequality.

506.

Elevators The total weight, w, of an elevator’s passengers can be no more than 1,200 pounds. Write this as an inequality.

507.

Shopping The number of items, n, a shopper can have in the express check-out lane is at most 8. Write this as an inequality.

### Writing Exercises

508.

Give an example from your life using the phrase ‘at least’.

509.

Give an example from your life using the phrase ‘at most’.

510.

Explain why it is necessary to reverse the inequality when solving $−5x>10−5x>10$.

511.

Explain why it is necessary to reverse the inequality when solving $n−3<12n−3<12$.

### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?