# How do you solve the absolute value of equation 3abs(2a+7)+6=3a+18?

Apr 2, 2015

We first make life a bit easier by dividing everything by $3$:

$| 2 a + 7 | + 2 = a + 6 \to | 2 a + 7 | = a + 4 \to$
$| 2 a + 7 | - a = 4$

(1)
For $a \ge - 3 \frac{1}{2}$ the absolutes are not necessary:

$2 a + 7 - a = 4 \to a = - 3$ which is acceptable

(2)
For $a < - 3 \frac{1}{2}$ the equation is turned around:

$- \left(2 a + 7\right) - a = 4 \to - 3 a = 11 \to a = - \frac{11}{3} = - 3 \frac{2}{3}$ which is below $- 3 \frac{1}{2}$ and so also acceptable

So the solutions are: $a = - 3$ and $a = - 3 \frac{2}{3}$
graph{|2x+7|-x [-12.54, 7.46, 1.16, 11.16]}