The absolute value function |x| always returns a non-negative number. It takes any number x and returns x if it's already non-negative, or -x if it is negative in order to make it positive.
[tex]|x|=\begin{cases}x&\text{for }x\ge0\\-x&\text{for }x<0\end{cases}[/tex]
For the equation
-3 + 4 |2x - 5| = 14
rearrange terms to get
|2x - 5| = 17/4
Now,
• if 2x - 5 ≥ 0, then |2x - 5| = 2x - 5. Then
2x - 5 = 17/4
• and if instead 2x - 5 < 0, then |2x - 5| = -(2x - 5), so that
-(2x - 5) = 17/4, or
2x - 5 = -17/4
In the first case,
2x - 5 = 17/4
2x = 17/4 + 5 = 37/4
x = 37/8
In the second case,
2x - 5 = -17/4
2x = -17/4 + 5 = 3/4
x = 3/8
