# 2.7E2: Exercises

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- David Arnold
- Retired Professor (Mathematics) at College of the Redwoods

## Practice Makes Perfect

**Graph Inequalities on the Number Line**

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

Exercise \(\PageIndex{1}\)

- \(x\leq 2\)
- x>−1
- x<0

Exercise \(\PageIndex{2}\)

- x>1
- x<−2
- \(x\geq −3\)

**Answer**-

Exercise \(\PageIndex{3}\)

- \(x\geq −3\)
- x<4
- \(x\leq −2\)

Exercise \(\PageIndex{4}\)

- \(x\leq 0\)
- x>−4
- \(x\geq −1\)

**Answer**-

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

Exercise \(\PageIndex{5}\)

- x<−2
- \(x\geq −3.5\)
- \( x\leq \frac{2}{3}\)

Exercise \(\PageIndex{6}\)

- \(x>3\)
- \(x \leq-0.5\)
- \(x \geq \frac{1}{3}\)

**Answer**-

Exercise \(\PageIndex{7}\)

- \(x \geq-4\)
- x<2.5
- \(x>-\frac{3}{2}\)

Exercise \(\PageIndex{8}\)

- \(x\leq 5\)
- \(x\geq −1.5x\)
- x<−73

**Answer**-

**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.

Exercise \(\PageIndex{9}\)

\(n-11<33\)

Exercise \(\PageIndex{10}\)

\(m-45 \leq 62\)

**Answer**

Exercise \(\PageIndex{11}\)

\(u+25>21\)

Exercise \(\PageIndex{12}\)

\(v+12>3\)

**Answer**

Exercise \(\PageIndex{13}\)

\(a+\frac{3}{4} \geq \frac{7}{10}\)

Exercise \(\PageIndex{14}\)

\(b+\frac{7}{8} \geq \frac{1}{6}\)

**Answer**

Exercise \(\PageIndex{15}\)

\(f-\frac{13}{20}<-\frac{5}{12}\)

Exercise \(\PageIndex{16}\)

\(g-\frac{11}{12}<-\frac{5}{18}\)

**Answer**

**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.

Exercise \(\PageIndex{17}\)

\(8 x>72\)

Exercise \(\PageIndex{18}\)

\(6 y<48\)

**Answer**

Exercise \(\PageIndex{19}\)

\(7 r \leq 56\)

Exercise \(\PageIndex{20}\)

\(9 s \geq 81\)

**Answer**

Exercise \(\PageIndex{21}\)

\(-5 u \geq 65\)

Exercise \(\PageIndex{22}\)

\(-8 v \leq 96\)

**Answer**

Exercise \(\PageIndex{23}\)

\(-9 c<126\)

Exercise \(\PageIndex{24}\)

\(-7 d>105\)

**Answer**

Exercise \(\PageIndex{25}\)

\(20>\frac{2}{5} h\)

Exercise \(\PageIndex{26}\)

\(40<\frac{5}{8} k\)

**Answer**

Exercise \(\PageIndex{27}\)

\(\frac{7}{6} j \geq 42\)

Exercise \(\PageIndex{28}\)

\(\frac{9}{4} g \leq 36\)

**Answer**

Exercise \(\PageIndex{29}\)

\(\frac{a}{-3} \leq 9\)

Exercise \(\PageIndex{30}\)

\(\frac{b}{-10} \geq 30\)

**Answer**

Exercise \(\PageIndex{31}\)

\(-25<\frac{p}{-5}\)

Exercise \(\PageIndex{32}\)

\(-18>\frac{q}{-6}\)

**Answer**

Exercise \(\PageIndex{33}\)

\(9 t \geq-27\)

Exercise \(\PageIndex{34}\)

\(7 s<-28\)

**Answer**

Exercise \(\PageIndex{35}\)

\(\frac{2}{3} y>-36\)

Exercise \(\PageIndex{36}\)

\(\frac{3}{5} x \leq-45\)

**Answer**

**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.

Exercise \(\PageIndex{37}\)

\(4 v \geq 9 v-40\)

Exercise \(\PageIndex{38}\)

\(5 u \leq 8 u-21\)

**Answer**

Exercise \(\PageIndex{39}\)

\(13 q<7 q-29\)

Exercise \(\PageIndex{40}\)

\(9 p>14 p-18\)

**Answer**

Exercise \(\PageIndex{41}\)

\(12 x+3(x+7)>10 x-24\)

Exercise \(\PageIndex{42}\)

\(9 y+5(y+3)<4 y-35\)

**Answer**

Exercise \(\PageIndex{43}\)

\(6 h-4(h-1) \leq 7 h-11\)

Exercise \(\PageIndex{44}\)

\(4 k-(k-2) \geq 7 k-26\)

**Answer**

Exercise \(\PageIndex{45}\)

\(8 m-2(14-m) \geq 7(m-4)+3 m\)

Exercise \(\PageIndex{46}\)

\(6 n-12(3-n) \leq 9(n-4)+9 n\)

**Answer**

Exercise \(\PageIndex{47}\)

\(\frac{3}{4} b-\frac{1}{3} b<\frac{5}{12} b-\frac{1}{2}\)

Exercise \(\PageIndex{48}\)

\(9 u+5(2 u-5) \geq 12(u-1)+7 u\)

**Answer**

Exercise \(\PageIndex{49}\)

\(\frac{2}{3} g-\frac{1}{2}(g-14) \leq \frac{1}{6}(g+42)\)

Exercise \(\PageIndex{50}\)

\(\frac{5}{6} a-\frac{1}{4} a>\frac{7}{12} a+\frac{2}{3}\)

**Answer**

Exercise \(\PageIndex{51}\)

\(\frac{4}{5} h-\frac{2}{3}(h-9) \geq \frac{1}{15}(2 h+90)\)

Exercise \(\PageIndex{52}\)

\(12 v+3(4 v-1) \leq 19(v-2)+5 v\)

**Answer**

**Mixed practice**

Exercise \(\PageIndex{53}\)

\(15 k \leq-40\)

Exercise \(\PageIndex{54}\)

\(35 k \geq-77\)

**Answer**

Exercise \(\PageIndex{55}\)

\(23 p-2(6-5 p)>3(11 p-4)\)

Exercise \(\PageIndex{56}\)

\(18 q-4(10-3 q)<5(6 q-8)\)

**Answer**

Exercise \(\PageIndex{57}\)

\(-\frac{9}{4} x \geq-\frac{5}{12}\)

Exercise \(\PageIndex{58}\)

\(-\frac{21}{8} y \leq-\frac{15}{28}\)

**Answer**

Exercise \(\PageIndex{59}\)

\(c+34<-99\)

Exercise \(\PageIndex{60}\)

\(d+29>-61\)

**Answer**

Exercise \(\PageIndex{61}\)

\(\frac{m}{18} \geq-4\)

Exercise \(\PageIndex{62}\)

\(\frac{n}{13} \leq-6\)

**Answer**

**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.

Exercise \(\PageIndex{63}\)

Fourteen times *d* is greater than 56.

Exercise \(\PageIndex{64}\)

Ninety times *c* is less than 450.

**Answer**

Exercise \(\PageIndex{65}\)

Eight times \(z\) is smaller than \(-40\)

Exercise \(\PageIndex{66}\)

Ten times \(y\) is at most \(-110\)

**Answer**

Exercise \(\PageIndex{67}\)

Three more than \(h\) is no less than 25

Exercise \(\PageIndex{68}\)

Six more than \(k\) exceeds 25

**Answer**

Exercise \(\PageIndex{69}\)

Ten less than \(w\) is at least \(39 .\)

Exercise \(\PageIndex{70}\)

Twelve less than \(x\) is no less than 21

**Answer**

Exercise \(\PageIndex{71}\)

Negative five times \(r\) is no more than \(95 .\)

Exercise \(\PageIndex{72}\)

Negative two times s is lower than 56

**Answer**

Exercise \(\PageIndex{73}\)

Nineteen less than \(b\) is at most \(-22\)

Exercise \(\PageIndex{74}\)

Fifteen less than a is at least \(-7\)

**Answer**

### Everyday Math

Exercise \(\PageIndex{75}\)

**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.

Exercise \(\PageIndex{76}\)

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

**Answer**-
\(h \leq 77\)

Exercise \(\PageIndex{77}\)

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

Exercise \(\PageIndex{78}\)

**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.

**Answer**-
\(n \leq 8\)

### Writing Exercises

Exercise \(\PageIndex{79}\)

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

Exercise \(\PageIndex{80}\)

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

**Answer**-
Answers will vary.

Exercise \(\PageIndex{81}\)

Explain why it is necessary to reverse the inequality when solving \(-5 x>10\)

Exercise \(\PageIndex{82}\)

Explain why it is necessary to reverse the inequality when solving \(\frac{n}{-3}<12\)

**Answer**-
Answers will vary.

### 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?