# How do you solve 3abs(w/2+5)< 18?

Apr 22, 2015

$3 \left\mid \frac{w}{2} + 5 \right\mid < 18$
could be written
$\left\mid \frac{w}{2} + 5 \right\mid < 6$

Two cases need to be considered

Case 1:
$\left(\frac{w}{2} + 5\right) < 0$
$\rightarrow w < - 10$

$\left\mid \frac{w}{2} + 5 \right\mid < 6$
becomes
$- \frac{w}{2} - 5 < 6$

$- \frac{w}{2} < 11$

$w > - 22$

Case 1 gives us
$w < - 10$ and $w > - 22$
$w \epsilon \left(- 22 , - 10\right)$

Case 2:
$\left(\frac{w}{2} + 5\right) \ge 0$
$\rightarrow w \ge - 10$

$\left\mid \frac{w}{2} + 5 \right\mid < 6$
becomes
$\frac{w}{2} + 5 < 6$

$\frac{w}{2} < 1$
$w < 2$

Case 2 gives us
$w \ge - 10$ and $w < 2$
$w \epsilon \left[- 10 , + 2\right)$

Union of the two cases:
$w \epsilon \left(- 22 , - 10\right) \cup w \epsilon \left[- 10 , + 2\right)$
gives a final solution

$w \epsilon \left(- 22 , + 2\right)$