# How do you solve -5|-2x + -3| + 2 = -33?

Jun 12, 2017

Given: $- 5 | - 2 x - 3 | + 2 = - 33$

Subtract 2 from both sides:

$- 5 | - 2 x + - 3 | = - 35$

Divide both sides by -5:

$| - 2 x + - 3 | = 7 \text{ [1]}$

Using the definition of the absolute value function,

|A| = {(A; A>=0),(-A;A < 0):}

We can write the following:

|-2x -3| = {(-2x-3; -2x-3>= 0),(2x+3;-2x-3<0):}

Simplify the inequalities:

|-2x -3| = {(-2x-3; -2x>= 3),(2x+3;-2x<3):}

|-2x -3| = {(-2x-3; x<= -3/2),(2x+3;x> -3/2):}

Substitute both into equation [1]:

-2x -3 = 7; x <= -3/2 and 2x + 3 = 7; x > -3/2

-2x = 10; x <= -3/2 and 2x = 4; x > -3/2

x = -5; x <= -3/2 and x = 2; x > -3/2

Check in the original equation:

$- 5 | - 2 \left(- 5\right) - 3 | + 2 = - 33$ and $- 5 | - 2 \left(2\right) - 3 | + 2 = - 33$

$- 5 | 7 | + 2 = - 33$ and $- 5 | - 7 | + 2 = - 33$

$- 33 = - 33$ and $- 33 = - 33$

Both values check.

$x = - 5$ and $x = 2$