*Yuanxin (Amy) Yang Alcocer*Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*
Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

Absolute value inequalities are types of problems that may seem complicated from the outset but are relatively simple to solve with the right understanding. Take a closer look at the definitions of the absolute value and the absolute value inequality, followed by an example of how to set up and solve these types of problems.
Updated: 09/30/2021

Before we talk about what an absolute value inequality is, let's talk about what an absolute value is. When we take the **absolute value** of something, we are trying to find the distance that value is from 0. For example, the absolute value of 5 is 5 since it is 5 away from 0. How about the absolute value of -5? What do you think that is? It is 5 as well because -5 is also 5 away from 0. Since we're talking math here, we also have a symbol for the absolute value and that is a vertical line on either side of the value as in |5| and |-5|.

We can solve these absolute values easily since we know that both are 5 away from 0. So, we know that |5| = 5 and |-5| = 5. If our problem was |*x*| = 5, then we already know that our answer is 5 and -5. Here's a trick question: Can we have |*x*| = -5? What do you think? Is it possible that the absolute value of something could equal a negative number? No, it's not, because we don't have a negative distance. So, if you ever see an absolute value equaling a negative number, you can right away say that the problem is false and can't be solved since the absolute value of anything can never be negative.

Now let's talk about absolute value inequalities.

An **absolute value inequality** is a problem with absolute values as well as inequality signs. We can have problems like |*x* + 3| > 1. We have four different inequality signs to choose from. We have less than, greater than, less than or equal to and greater than or equal to. So, our absolute value inequalities can have any one of these four signs.

The process to solve them is the same regardless of the sign. But you do have to pay attention to what type of sign it is, as you will be taking that into account when you solve. Just like our regular absolute values, we can't have our absolute value being a negative number. So, a problem such as |*x* + 3| < 0 or less than a negative number will not be possible and cannot be solved.

Now let's see about solving an absolute value inequality. We will work with the inequality |*x* + 3| > 1. To set up our problem for solving, we need to write two problems from our one absolute value inequality. We do this by writing one problem exactly as we see the absolute value inequality but without the absolute value signs. The next one we will write will be almost the same except we are flipping the inequality sign around and changing the right side to the negative version.

So, for our absolute value inequality, our two problems are *x* + 3 > 1 and *x* + 3 < -1. Notice how I've flipped the inequality around in the second problem, and I've changed my right side to the negative version. If the right side is already negative, then we write the positive since the negative of a negative is a positive. Once we've done this, we are ready to proceed and find our answers.

Looking at our two problems, we see that all that is needed to solve them is a bit of algebra. All we need to do is to move the three over to the other side. We can do that easily for both by subtracting 3 from both sides. Doing this to both problems we get *x* > -2 and *x* < -4. It might look like we're done, but we aren't done just yet. We need to write in proper answer format. Because our sign is greater than, our proper answer is *x* > -2 OR *x* < -4. We need to have an OR in between our two answers. This applies when we have greater than or greater than or equal to symbols. If we had less than or less than or equal to symbols, then we would have an AND in between our answers.

Another thing to note is that when you need to multiply or divide by a negative number to get x by itself, you also need to flip the inequality sign. For example, to solve -2*x* < 4, we need to divide by a -2. Because we are dividing by a negative number, we need to flip the inequality sign around, so our answer becomes *x* > -2. Notice how our sign has flipped, but everything else remains the same.

Let's recap what we have learned. We've learned that an **absolute value** is the distance from 0. The math symbol used for absolute values is a set of straight vertical lines on either side of the value, like |*x*|. Our absolute values can never be negative, so problems such as |*x*| = -4 cannot be solved. An **absolute value inequality** is an absolute value problem with inequalities. The same restriction applies about the absolute value not being negative. We can have problems such as |3*x* - 5| < 4, but we can't have problems such as |3*x* - 5| < -3.

To solve an absolute value inequality, we set up two problems from our one absolute value inequality. Our first problem is the same as our inequality except it doesn't have the absolute value symbols. The next problem likewise doesn't have the absolute value symbols, but the inequality symbol is flipped and the right side has been negated. Once we have set up our two problems, we go ahead with algebra to solve for our variable. If our inequality symbol is greater than or greater than or equal to, then our two answers will be separated by an OR. If the inequality symbol is less than or less than or equal to, then our two answers will be separated by an AND.

Your focus on this video lesson could allow you to:

- Identify an absolute value equation and solve it
- Illustrate the absolute value symbol
- Understand and solve for an absolute value inequality

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