# Chapter 6: Linear Inequalities and Absolute Value

**Basic**Created by: CK-12

## Introduction

This chapter moves beyond equations to the study of inequalities. Many situations have more than one correct answer. A police officer can issue a ticket for any speed exceeding the limit. A rider for the bumper boats must be less than 48 inches tall. Both these situations have many possible answers.

- 6.1.
## Inequality Expressions

- 6.2.
## Inequalities with Addition and Subtraction

- 6.3.
## Inequalities with Multiplication and Division

- 6.4.
## Multi-Step Inequalities

- 6.5.
## Compound Inequalities

- 6.6.
## Applications with Inequalities

- 6.7.
## Absolute Value

- 6.8.
## Absolute Value Equations

- 6.9.
## Graphs of Absolute Value Equations

- 6.10.
## Absolute Value Inequalities

- 6.11.
## Linear Inequalities in Two Variables

- 6.12.
## Theoretical and Experimental Probability

### Chapter Summary

## Summary

This chapter begins by talking about inequalities and how to solve them by using addition, subtraction, multiplication, and division. Multi-step and compound inequalities are covered, as are real-world inequalities. In addition, linear inequalities in two variables are discussed. The chapter then moves on to absolute values, giving instruction on how to solve and graph absolute value equations and inequalities. Finally, the chapter wraps up by exploring theoretical and experimental probability.

### Linear Inequalities and Absolute Value; An Introduction to Probability Review

Vocabulary: In 1–12, define the term.

- Algebraic inequality
- Interval notation
- Intersection of sets
- Union of sets
- Absolute value
- Compound inequality
- Boundary line
- Half plane
- Solution set
- Probability
- Theoretical probability
- Experimental probability
- Find the distance between 16 and 104 on a number line.
- Shanna needed less than one dozen eggs to bake a cake. Write this situation as an inequality and graph the appropriate solutions on a number line.
- Yemi can walk no more than 8 dogs at once. Write this situation as an inequality and graph the appropriate solutions on a number line.

In 16–35, solve each inequality. Graph the solutions on a number line.

- \begin{align*}y+7 \ge 36\end{align*}
y+7≥36 - \begin{align*}16x<1\end{align*}
16x<1 - \begin{align*}y-64<-64\end{align*}
y−64<−64 - \begin{align*}5> \frac{t}{3}\end{align*}
5>t3 - \begin{align*}0 \le 6-k\end{align*}
0≤6−k - \begin{align*}-\frac{3}{4} g \le 12\end{align*}
−34g≤12 - \begin{align*}10 \ge \frac{q}{-3}\end{align*}
10≥q−3 - \begin{align*}-14+m>7\end{align*}
−14+m>7 - \begin{align*}4 \ge d+11\end{align*}
4≥d+11 - \begin{align*}t-9 \le -100\end{align*}
t−9≤−100 - \begin{align*}\frac{v}{7}<-2\end{align*}
v7<−2 - \begin{align*}4x \ge -4\end{align*}
4x≥−4 and \begin{align*}\frac{x}{5}<0\end{align*}x5<0 - \begin{align*}n-1 < -5\end{align*}
n−1<−5 or \begin{align*}\frac{n}{3}\ge -1\end{align*}n3≥−1 - \begin{align*}\frac{n}{2}>-2\end{align*}
n2>−2 and \begin{align*}-5n > -20\end{align*}−5n>−20 - \begin{align*}-35 + 3x > 5(x-5)\end{align*}
−35+3x>5(x−5) - \begin{align*}x+6-11x \ge -2(3+5x)+12(x+12)\end{align*}
x+6−11x≥−2(3+5x)+12(x+12) - \begin{align*}-64 < 8(6+2k)\end{align*}
−64<8(6+2k) - \begin{align*}0 > 2(x+4)\end{align*}
0>2(x+4) - \begin{align*}-4(2n-7) \le 37-5n\end{align*}
−4(2n−7)≤37−5n - \begin{align*}6b+14 \le -8(-5b-6)\end{align*}
6b+14≤−8(−5b−6) - How many solutions does the inequality \begin{align*}6b+14 \le -8(-5b-6)\end{align*}
6b+14≤−8(−5b−6) have? - How many solutions does the inequality \begin{align*}6x+11<3(2x-5)\end{align*}
6x+11<3(2x−5) have? - Terry wants to rent a car. The company he’s chosen charges $25 a day and $0.15 per mile. If he rents it for one day, how many miles would he have to drive to pay at least $108?
- Quality control can accept a part if it falls within \begin{align*}\pm\end{align*}
± 0.015 cm of the target length. The target length of the part is 15 cm. What is the range of values quality control can accept? - Strawberries cost $1.67 per pound and blueberries cost $1.89 per pound. Graph the possibilities that Shawna can buy with no more than $12.00.

Solve each absolute value equation.

- \begin{align*}24=|8z|\end{align*}
24=|8z| - \begin{align*}\left |\frac{u}{4}\right |=-1.5\end{align*}
∣∣u4∣∣=−1.5 - \begin{align*}1=|4r-7|-2\end{align*}
1=|4r−7|−2 - \begin{align*}|-9+x|=7\end{align*}
|−9+x|=7

Graph each inequality or equation.

- \begin{align*}y=|x|-2\end{align*}
y=|x|−2 - \begin{align*}y=-|x+4|\end{align*}
y=−|x+4| - \begin{align*}y=|x+1|+1\end{align*}
- \begin{align*}y \ge -x+3\end{align*}
- \begin{align*}y<-3x+7\end{align*}
- \begin{align*}3x+y \le -4\end{align*}
- \begin{align*}y>\frac{-1}{4} x+6\end{align*}
- \begin{align*}8x-3y\le -12\end{align*}
- \begin{align*}x<-3\end{align*}
- \begin{align*}y>-5\end{align*}
- \begin{align*}-2<x\le 5\end{align*}
- \begin{align*}0\le y \le 3\end{align*}
- \begin{align*}|x|>4\end{align*}
- \begin{align*}|y|\le -2\end{align*}

A spinner is divided into eight equally spaced sections, numbered 1 through 8. Use this information to answer the following questions.

- Write the sample space for this experiment.
- What is the theoretical probability of the spinner landing on 7?
- Give the probability that the spinner lands on an even number.
- What are the odds for landing on a multiple of 2?
- What are the odds against landing on a prime number?
- Use the TI Probability Simulator application “Spinner.” Create an identical spinner. Perform the experiment 15 times. What is the experimental probability of landing on a 3?
- What is the probability of the spinner landing on a number greater than 5?
- Give an event with a 100% probability.
- Give an event with a 50% probability.

### Linear Inequalities and Absolute Value; An Introduction to Probability Test

- Consider a standard 52-card deck. Determine:
- \begin{align*}P\end{align*}(
*red*4) - \begin{align*}P\end{align*}(
*purple Ace*) - \begin{align*}P\end{align*}(
*number card*)

- \begin{align*}P\end{align*}(
- Solve \begin{align*}-7 \le y+7<5\end{align*}.
- Find the distance between –1.5 and 9.
- Solve \begin{align*}23=|8-7r|+3\end{align*}.
- Solve \begin{align*}|-7c| \ge 49\end{align*}.
- Graph \begin{align*}x-2y \le 10\end{align*}.
- Graph \begin{align*}y>-\frac{3}{5} x+4\end{align*}.
- Graph \begin{align*}y=-|x-3|\end{align*}.
- A bag contains 2 red socks, 3 blue socks, and 4 black socks.
- If you choose one sock at a time, write the sample space.
- Find \begin{align*}P\end{align*}(
*blue sock*). - Find the odds against drawing a black sock.
- Find the odds for drawing a red sock.

Solve each inequality.

- \begin{align*}2(6+7r)>-12+8r\end{align*}
- \begin{align*}-56 \le 8 + 8(7x+6)\end{align*}

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