Question

Which of the following are solutions to | x - 1 |=5x + 2 ?

Answer 1

Use the definition of absolute value,

$|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$

If x - 1 ≥ 0, then |x - 1| = x - 1, and so

|x - 1| = 5x + 2

x - 1 = 5x + 2

4x = -3

x = -3/4

Otherwise, if x - 1 < 0, then |x - 1| = -(x - 1), which means

|x - 1| = 5x + 2

-x + 1 = 5x + 2

6x = -1

x = -1/6