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

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

If `|f(x)| < n`, then

`-n < f(x) < n`

Example 1

Solve the inequality |x − 3| < 2.

Example 2

Solve the inequality |2x − 1| > 5

Example 3

Solve the inequality `2|(2x)/3 + 1|>=4`

Example 4

Solve the inequality `|3 − 2x| < 3`

Example 5

A technician measures an electric current which is `0.036\ "A"` with a possible error of `±0.002\ "A"`. Write this current, i, as an inequality with absolute values.

Exercises

1. Solve ` |5 − x| ≤ 2`

2. Solve `|(2x-9)/4|<1`

3. Solve `|(4x)/3-5|>=7`

4. Solve `|x^2+3x-1|<3`