You may want to review absolute-value inequalities and solving absolute-value equations first.

Absolute-value inequalities require you to pull together lots of things you've learned. Suppose you have |x − 2| − 3 ≤ 2, First, you have to get the absolute value by itself on one side. Now we have |x − 2| ≤ 5.

The tricky part is that now we have to solve the equation |x − 2| = 5. Its solution is {7, −3}. But that's just the solution of the equation, not of the inequality. To solve the inequality, we need to put those numbers into either an interval or a union of intervals, using the table here, only backwards. Since our inequality had a ‘≤’ sign, the answer is [−3, 7]. Note that the smaller number always comes first. With braces, that doesn't matter, but with intervals it does.

Sometimes (where the inequality has a ‘>’ or ‘≥’), you'll need to write the union of two intervals. That's why I left two spaces for you, with a ‘∪’ between them. If you only need one, just leave the second blank.

Once you've gotten the absolute value by itself, you should always check that the right-hand side isn't negative. If it's negative, then it won't match any pattern in the table. In that case, if the inequality has a ‘≤’ or a ‘<’, it's a contradiction. On this website, you can say so by leaving both spots for the answer blank. (Your teacher might want you to say so some other way.) If the inequality has a ‘≥’ or a ‘>’, then it's an identity. Since that means it's right no matter what x is, you can write that as the interval (−∞, ∞). (If you decide it's a contradiction or an identity for some other reason, you should also say so the same way.)