### About Absolute Value Equations Part 2:

In some cases, we will face absolute value equations with quadratic expressions and rational expressions involved. In order to solve these types of equations, we turn to our rule: if |u| = a, then u = a or u = -a.

Test Objectives
• Demonstrate the ability to solve an absolute value equation with a quadratic expression
• Demonstrate the ability to solve an absolute value equation with a rational expression
Absolute Value Equations Part 2 Practice Test:

#1:

Instructions: solve each equation.

$$a)\hspace{.2em}2|x^2 + 8x - 4| - 5=5$$

$$b)\hspace{.2em}-2|2x^2 - 7x + 5| + 3=-7$$

#2:

Instructions: solve each equation.

$$a)\hspace{.2em}\frac{1}{2}|3x^2 - 14x - 1| + 1=\frac{41}{2}$$

#3:

Instructions: solve each equation.

$$a)\hspace{.2em}|3x^2 - 8|=|x - 2|$$

#4:

Instructions: solve each equation.

$$a)\hspace{.2em}\left|\frac{10}{x - 1}\right|=\left|\frac{5}{x + 3}\right|$$

#5:

Instructions: solve each equation.

$$a)\hspace{.2em}\left|\frac{1}{2x - 5}\right|=\left|\frac{3}{x - 7}\right|$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}x=-9, -4 \pm \sqrt{15}, 1$$

$$b)\hspace{.2em}x=0, \frac{7}{2}, \frac{7 \pm i \sqrt{31}}{4}$$

#2:

Solutions:

$$a)\hspace{.2em}x=-2, \frac{20}{3}, \frac{7 \pm i \sqrt{65}}{3}$$

#3:

Solutions:

$$a)\hspace{.2em}x=-2, \frac{5}{3}, \frac{1 \pm \sqrt{73}}{6}$$

#4:

Solutions:

$$a)\hspace{.2em}x=-7, -\frac{5}{3}$$

#5:

Solutions:

$$a)\hspace{.2em}x=\frac{8}{5}, \frac{22}{7}$$