Solving Absolute Value Inequalities

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

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.

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

Example 1

\(|3x-4|\)< \(7\)

 

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

\(-7\)< \(3x – 4\) < \(7\)

 

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

\(-3\) < \(3x\) < \(11\)

 

Now, divide by 3 to get \(x\) by itself.

\(-1\) < \(x\) < \(\frac{11}{3}\)

 

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.

Let’s try another one.

Example 2

\(|-2x + 3|\) ≤ \(9\)

 

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

\(-9\) ≤ \(-2x + 3\) ≤ \(9\)

 

Then, subtract 3 from each part.

\(-12\) ≤ \(-2x\) ≤ \(6\)

 

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

\(6\) ≥ \(x\) ≥ \(-3\)

 

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.

\(|-5x – 10|\) ≤ \(15\)

 

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

\(-15\) ≤ \(-5x – 10\) ≤ \(15\)

 

Then, add 10 to all 3 parts.

\(-5\) ≤ \(-5x\) ≤ \(25\)

 

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

\(1\) ≥ \(x\) ≥ \(-5\)

 

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!

Practice Questions

Question #1:

 
\(|5x+14|\leq9\)

\(-\frac{14}{5}\leq x \leq\frac{9}{5}\)
\(9\leq x \leq9\)
\(-23\leq x \leq-5\)
\(-\frac{23}{5}\leq x \leq-1\)
Answer:

The correct answer is D. The first thing to do when solving absolute value inequalities is place the expression between positive and negative values of the given numbers, like this:
\(-9\leq 5x+14 \leq 9\)

Then, subtract 14 from all three parts.
\(-23\leq5x\leq-5\)

Finally, divide each part by 5.
\(-\frac{23}{5}\leq x \leq-1\)

Question #2:

 
\(|7x-11|\)<\(12\)

\(-\frac{1}{7}\)<\(x\)<\(\frac{23}{7}\)

\(-12\)<\(x\)<\(7\)

\(-1\)<\(x\)<\(23\)

\(-\frac{4}{7}\)<\(x\)<\(\frac{16}{7}\)

Answer:

The correct answer is A. The first step in solving absolute value inequalities is to place the expression between positive and negative values of the given number and get rid of the absolute value symbols, like this:
\(-12\)<\(7x-11\)<\(12\)

Then, add 11 to all three parts.
\(-1\)<\(7x\)<\(23\)

Finally, divide each part by 7.
\(-\frac{1}{7}\)<\(x\)<\(\frac{23}{7}\)

Question #3:

 
\(|7x+3|-2\leq19\)

\(-24\leq x \leq 18\)
\(\frac{11}{7}\leq x \leq \frac{18}{7}\)
\(-\frac{24}{7}\leq x \leq \frac{18}{7}\)
\(-\frac{18}{7}\leq x \leq \frac{24}{7}\)
Answer:

The correct answer is C. The first step in this problem is to isolate the absolute value expression on one side of the inequality. We can do this by adding 2 to both sides.
\(|7x+3|\leq21\)

Then, to get rid of the absolute value, place \(7x+3\) between the ±21, like so:
\(-21\leq 7x+3 \leq21\)

Next, subtract 3 from all 3 parts.
\(-24\leq7 x \leq18\)

Finally, divide each part by 7.
\(-\frac{24}{7}\leq x \leq \frac{18}{7}\)

Question #4:

 
\(\frac{1}{2}|6x-4|\leq17\)

\(-30\leq x \leq 38\)
\(-5\leq x \leq \frac{19}{3}\)
\(-30\leq x \leq \frac{19}{3}\)
\(-5\leq x \leq 38\)
Answer:

The correct answer is B. First, isolate the absolute value expression on one side of the inequality. In this case, do that by multiplying both sides by 2.
\(|6x-4|\leq34\)

Then, get rid of the absolute value signs by placing \(6x-4\) between \(\pm34\), like this:
\(-34\leq 6x-4 \leq 34\)

Next, add 4 to all three parts.
\(-30\leq 6x \leq 38\)

Finally, divided each part by 6, being sure to simplify fractions when necessary.
\(-5\leq x \leq \frac{19}{3}\)

Question #5:

 
\(3|2x-5|+12\)<\(39\)

\(-14\)<\(x\)<\(4\)

\(-4\)<\(x\)<\(14\)

\(-7\)<\(x\)<\(2\)

\(-2\)<\(x\)<\(7\)

Answer:

The correct answer is D. The first thing to do in order to solve this inequality is isolate the absolute value expression on one side of the inequality. Do this by first subtracting 12 from both sides, then divide both sides by 3.
\(3|2x-5|\)<\(27\) \(|2x-5|\)<\(9\)

Then, get rid of the absolute value signs by putting the expression \(2x-5\) between \(\pm9\), like this:
\(-9\)<\(2x-5\)<\(9\)

Next, add 5 to all three parts.
\(-4\)<\(2x\)<\(14\)

Finally, divide each part by 2.
\(-2\)<\(x\)<\(7\)

 

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by Mometrix Test Preparation | This Page Last Updated: June 14, 2022