Loading [MathJax]/jax/output/HTML-CSS/fonts/TeX/fontdata.js

Solving Absolute Value Inequalities

Solving Absolute Value Inequalities Video

Hi, and welcome to this video on solving absolute value inequalities!

What is an Absolute Value Inequality?

An absolute value inequality is an inequality that has an absolute value on one side of the inequality.=

Remember, when you solve an absolute value equation, you come up with two answers. An absolute value inequality is similar, except instead of two answers, your answer will include all the numbers between the two that you found.

Solving Absolute Value Inequalities

To solve an absolute value inequality, remove the absolute value signs, and place the expression between the positive and negative values of the inequality given to you. Then, solve the problem like you would any other inequality expression, remembering to do the same thing to all three parts of the expression.

Example #1

Let’s look at an example to see what I’m talking about.

|3x4|< 7

 
First, we remove the absolute value signs and place the expression between positive and negative 7, like this:

7< 3x4 < 7

 
Then, we solve for x. First, we add 4 to each part.

3 < 3x < 11

 
Now, divide by 3 to get x by itself.

1 < x < 113

 
And that’s your answer. Notice how you still solve for two different numbers, but your answer is the range of numbers between those two.

Example #2

Let’s try another one.

|2x+3|9

 
First, get rid of the absolute value signs, and put the expression between -9 and 9.

92x+39

 
Then, subtract 3 from each part.

122x6

 

And finally, divide by -2. But remember, when you divide by a negative, you have to flip the inequality signs.

6x3

 

And there’s your answer!

Example #3

I want you to try one more, but this time pause the video and try to figure it out yourself. Then, check your steps with mine.

|5x10|15

 
First, remove the absolute value signs and place your expression between -15 and 15.

155x1015

 
Then, add 10 to all 3 parts.

55x25

 
Divide by -5, remembering to flip your signs since you’re dividing by a negative.

1x5

 
And that’s all there is to it! I hope this video on solving absolute value inequalities was helpful. Thanks for watching, and happy studying!

Absolute Value Inequalities Practice Questions

Question #1:

 
|5x+14|9

145x95
9x9
23x5
235x1
Question #2:

 
|7x11|<12

17<x<237
12<x<7
1<x<23
47<x<167
Question #3:

 
|7x+3|219

24x18
117x187
247x187
187x247
Question #4:

 
12|6x4|17

30x38
5x193
30x193
5x38
Question #5:

 
3|2x5|+12<39

14<x<4
4<x<14
7<x<2
2<x<7
997008451494

 

by Mometrix Test Preparation | Last Updated: February 10, 2025