#### Answer

$x= \frac{-6}{5}$

#### Work Step by Step

$|x-2| = 4x + 8 $
There are two cases.
Case 1: $x-2 = 4x + 8$
Case 2: $x-2= -(4x+8)$
Solving for $x$ in Case 1:
Subtract $x$ from both sides of the equation.
$-2 = 3x + 8$
Subtract $8$ from both sides of the equation.
$-10 = 3x$
Divide both sides by $3$ to isolate $x$.
$x = \frac{-10}{3}$
However, if we plug in $\frac{-10}{3}$ as $x$ into the equation, we get $|\frac{-10}{3} -2| = 4(\frac{-10}{3}) + 8$
This simplifies to $ |\frac{-16}{3}| = \frac{-16}{3}$ , which is not possible, because absolute values cannot be negative.
Let's solve for $x$ in Case 2:
Subtract $x$ from both sides of the equation.
$-2 = -5x - 8$
Add $8$ to both sides.
$6 = -5x$
Divide both sides by $-5$.
$x = \frac{-6}{5} $
If we plug in $\frac{-6}{5}$ for $x$, we get
$| \frac{-6}{5} - 2| = 4(\frac{-6}{5}) + 8$
This simplifies to $|\frac{-16}{5}| = \frac{16}{5}$, which satisfies the equation.
Thus $x$ only has one solution, $\frac{-6}{5}$