Question

Solve the following absolute value equation for the unknown. Show all of your work for full credit. |-3h – 6| ≤ 3

Answer 1

`-3 \le h \le 1`.

Explanation:

Apply the property of absolute values: if
`a \ge 0`, then
`|x| \le a \iff -a \le x \le a`. By this property,
`|- 3\, h - 6 | \le 3` is equivalent to
`-3 \le -3\, h - 6\le 3`. That's the same as saying that
`-3\, h - 6 \ge -3` and
`-3\, h - 6 \le 3`.

Add
`6` to both sides of both inequalities:

`-3\, h \ge 3` and
`-3\, h \le 9`.

Divide both sides of both inequalities by
`(-3)`. Note that because
`-3 < 0`, dividing both sides of an equality by this number will flip the direction of the inequality sign.

• 
  `-3\, h \ge 3` would become
  `h \le -1`.
• 
  `-3\, h \le 9` would become
  `h \ge -3`.

Both inequalities are supposed to be true. Combining the two inequalities to obtain:

`-3 \le h \le 1`.