How do you solve abs(2/3x-1/4x)=abs(1/4x+8)?

Apr 3, 2015

$x = 48$ and $x = - 12$

Solution:
$| \frac{2}{3} x - \frac{1}{4} x | = | \frac{1}{4} x + 8 |$

$\implies | \frac{8 - 3}{12} x | = | \frac{1}{4} x + 8 |$

$\implies | \frac{5}{12} x | = | \frac{1}{4} x + 8 |$

Square both sides, you get

${\left(\frac{5}{12} x\right)}^{2} = {\left(\frac{1}{4} x + 8\right)}^{2}$

$\implies {\left(\frac{5}{12} x\right)}^{2} - {\left(\frac{1}{4} x + 8\right)}^{2} = 0$

This is a difference of two squares, as ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

$\implies \left(\frac{5}{12} x - \left(\frac{1}{4} x + 8\right)\right) \cdot \left(\frac{5}{12} x + \left(\frac{1}{4} x + 8\right)\right) = 0$

$\implies \left(\frac{2}{12} x - 8\right) \left(\frac{8}{12} x + 8\right) = 0$

$\implies \frac{2}{12} x - 8 = 0 \implies \frac{1}{6} x = 8 \implies x = 48$

Also,

$\frac{8}{12} x + 8 = 0 \implies \frac{2}{3} x = - 8 \implies x = - 12$