About Solving Absolute Value Inequalities:

When we solve absolute value inequalities, we revisit the concept of absolute value. To think about a scenario such as: |x| < 3, we find all numbers whose absolute value is less than 3. When we think about an alternative scenario such as: |x| > 3, we find all numbers whose absolute value is larger than 3.


Test Objectives
  • Demonstrate a general understanding of absolute value
  • Demonstrate the ability to solve a compound inequality with "and" or "or"
  • Demonstrate the ability to solve an absolute value inequality
Solving Absolute Value Inequalities Practice Test:

#1:

Instructions: Solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}|1 - 5x| ≥ 36$$

$$b)\hspace{.2em}|10 + 9x| < 73$$


#2:

Instructions: Solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}|-4 - 7x| ≤ 10$$

$$b)\hspace{.2em}|5x - 10| < 5$$


#3:

Instructions: Solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}|6 - 3x| > 3$$

$$b)\hspace{.2em}8|3x + 4| - 1 ≤ 55$$


#4:

Instructions: Solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}|3 + 10x| - 6 ≤ 41$$

$$b)\hspace{.2em}7|8x + 10| - 3 ≥ -45$$


#5:

Instructions: Solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}-|7x - 8| + 2 ≤ -32$$

$$b)\hspace{.2em}-6|7 - 6x| + 2 ≥ 68$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}x ≤ -7 \hspace{.2em}or \hspace{.2em}x ≥ \frac{37}{5}$$ $$(-\infty, -7] ∪ \left[\frac{37}{5}, \infty\right)$$

graphing our interval on the number line

$$b)\hspace{.2em}-\frac{83}{9}< x < 7$$ $$\left(-\frac{83}{9}, 7\right)$$

graphing our interval on the number line

#2:

Solutions:

$$a)\hspace{.2em}-2 ≤ x ≤ \frac{6}{7}$$ $$\left[-2, \frac{6}{7}\right]$$

graphing our interval on the number line

$$b)\hspace{.2em}1 < x < 3$$ $$(1,3)$$

graphing our interval on the number line

#3:

Solutions:

$$a)\hspace{.2em}x < 1 \hspace{.2em}or \hspace{.2em}x > 3$$ $$(-\infty, 1) ∪ (3, \infty)$$

graphing our interval on the number line

$$b)\hspace{.2em}-\frac{11}{3}≤ x ≤ 1$$ $$\left[-\frac{11}{3}, 1\right]$$

graphing our interval on the number line

#4:

Solutions:

$$a)\hspace{.2em}-5 ≤ x ≤ \frac{22}{5}$$ $$\left[-5, \frac{22}{5}\right]$$

graphing our interval on the number line

$$b)\hspace{.2em}All \hspace{.2em}Real \hspace{.2em}Numbers$$ $$(-\infty, \infty)$$

graphing our interval on the number line

#5:

Solutions:

$$a)\hspace{.2em}x ≤ -\frac{26}{7}\hspace{.2em}or \hspace{.2em}x ≥ 6$$ $$\left(-\infty, -\frac{26}{7}\right] ∪ [6, \infty)$$

graphing our interval on the number line

$$b)\hspace{.2em}No \hspace{.2em}Solution$$ $$∅$$