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

With absolute value signs, equations and inequalities become much more complicated! Here, you'll learn how to write an equation or inequality, if you're told what its solutions should be.

The solution you're given should involve two numbers, say 2 and 4. (We'll use those as our example.) Now we need to work out two things. The first of those is the number halfway in between 2 and 4. That's 3. The other is how far 2 and 4 are from that number 3 we just worked out. (They have to be the same distance, since 3 was halfway in between.) The answer is 1.

Now there are five possibilities for what the question could look like, and the answer will be different in each case. You'll want to memorize this table. (It will help you later.)

If you see: Then write:
{2, 4} |x − 3| = 1
(2, 4) |x − 3| < 1
[2, 4] |x − 3| ≤ 1
(−∞, 2) ∪ (4, ∞) |x − 3| > 1
(−∞, 2] ∪ [4, ∞) |x − 3| ≥ 1

Here are some things to keep in mind:

• The number on the right-hand side can never be negative.
• You might never have seen something like “{2, 4}” before. That just means x is either 2 or 4. That's different from (2, 4), which means x is between 2 and 4. It's also different from [2, 4], which means x is between 2 and 4, or is actually 2 or 4.
• On this website, I've put ‘|’ signs around the left-hand side for you, since they're always there. On pencil-and-paper tests, though, you need to be sure to write them.