Solve |x − 2| < 3
Absolute value inequality, worked out line by line the way a teacher would write it.
Answer
| Solution | −1 < x < 5 |
| Interval notation | (−1, 5) |
Step-by-step solution
4 steps-
1 Given\left|{x - 2}\right| < 3
Solve for x.
-
2 |u| < d means u lies between −d and d
|u| is the distance from u to 0. Being less than 3 away from 0 means u lies between −3 and 3.
-3 < x - 2 < 3 -
3 Add 2 to all three parts
A compound inequality is two inequalities at once. Doing the same operation to all three parts keeps both of them true, which isolates the variable in the middle.
-1 < x < 5 -
4 Solution-1 < x < 5\qquad \left( -1, 5 \right)
Check with test values
| x | Left side | Right side | Holds? |
|---|---|---|---|
| 0 | 2 | 3 | ✓ true |
| −2 | 4 | 3 | ✗ false |
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