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Video: Finding the Solution Set of an Absolute Value Inequality

Kathryn Kingham

Find the solution set of the inequality |x+4|< 9.

02:42

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, we’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, we’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.

Here’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.