# Question Video: Finding the Solution Set of an Absolute Value Inequality

Find the solution set of the inequality |π₯ + 4| < 9.

02:41

### Video Transcript

Find the solution set of the inequality, the absolute value of π₯ plus four is less than nine.

To solve the absolute value, weβll need to break this equation down into two pieces. We have the case when π₯ plus four is less than nine, the positive case. But we also have when negative π₯ plus four is less than nine.

We can start solving the problem on the left by distributing our negative. What we have now is negative π₯ minus four is less than nine. And then we treat this problem just like any other inequality. We isolate π₯. I can do that by adding four to both sides of the inequality. Negative π₯ minus four plus four leaves us with negative π₯. And nine plus four equals 13.

But what we have now is a negative π₯ and we need positive π₯. To do that, weβre going to multiply the entire inequality by negative one. Negative one times negative π₯ equals π₯. When dealing with inequalities, we multiply negative one times the inequality, and that means we flip the sign, π₯ is greater than. Multiplying negative one times 13, π₯ is greater than negative 13.

We look at our second inequality, π₯ plus four is less than nine. We can subtract four from both sides of the inequality, and we have π₯ is less than five.

Hereβs what we know, that π₯ is greater than negative 13 and π₯ is less than five, which means that our π₯ falls between negative 13 and five. We can write that in a compound inequality like this: π₯ falls between negative 13 and five. Another way to write this would be with brackets facing outward. And it says the same thing; π₯ is between negative 13 and five.

All of these three are ways to represent the solution to this inequality. Our answer choice is looking for a solution set like this: π₯ falls between negative 13 and five.