4. Solving Inequalities with Absolute Values

For inequalities involving absolute values ie. |x|, we use the following relationships, for some number n:

If `|f(x)| > n`, then this means:

`f(x) < -n` or `f(x) > n`

If `|f(x)| < n`, then this means:

`-n < f(x) < n`

Solve the inequality |x − 3| < 2.

Answer

Applying the relationships discussed above:

`- 2 < x - 3 < 2`

Adding `3` to all sides, we get:

`-2 + 3 < x - 3 + 3 < 2 + 3`

`1 < x < 5`

Here is the graph of our solution: