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. This translates into a three-part inequality: -3 < x < 3. When we think about an alternative scenario such as: |x| > 3, we find all numbers whose absolute value is larger than 3. This translates into a compound inequality with "or": x < -3 or x > 3. Additionally, we will come across many special case scenarios that require careful analysis. Always remember |a| ≥ 0 for all real numbers a.
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
#1:
Instructions: Solve each inequality, write in interval notation, graph.
$$a)\hspace{.2em}|1 - 5x| ≥ 36$$
$$b)\hspace{.2em}|10 + 9x| < 73$$
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#2:
Instructions: Solve each inequality, write in interval notation, graph.
$$a)\hspace{.2em}|{-}4 - 7x| ≤ 10$$
$$b)\hspace{.2em}|5x - 10| < 5$$
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#3:
Instructions: Solve each inequality, write in interval notation, graph.
$$a)\hspace{.2em}|6 - 3x| > 3$$
$$b)\hspace{.2em}8|3x + 4| - 1 ≤ 55$$
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#4:
Instructions: Solve each inequality, write in interval notation, graph.
$$a)\hspace{.2em}|3 + 10x| - 6 ≤ 41$$
$$b)\hspace{.2em}7|8x + 10| - 3 ≥ -45$$
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#5:
Instructions: Solve each inequality, write in interval notation, graph.
$$a)\hspace{.2em}{-}|7x - 8| + 2 ≤ -32$$
$$b)\hspace{.2em}{-}6|7 - 6x| + 2 ≥ 68$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}x ≤ -7 \hspace{.2em}\text{or} \hspace{.2em}x ≥ \frac{37}{5}$$ $$(-\infty, -7] ∪ \left[\frac{37}{5}, \infty\right)$$ 
$$b)\hspace{.2em}{-}\frac{83}{9}< x < 7$$ $$\left(-\frac{83}{9}, 7\right)$$ 
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#2:
Solutions:
$$a)\hspace{.2em}{-}2 ≤ x ≤ \frac{6}{7}$$ $$\left[-2, \frac{6}{7}\right]$$ 
$$b)\hspace{.2em}1 < x < 3$$ $$(1,3)$$ 
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#3:
Solutions:
$$a)\hspace{.2em}x < 1 \hspace{.2em}\text{or} \hspace{.2em}x > 3$$ $$(-\infty, 1) ∪ (3, \infty)$$ 
$$b)\hspace{.2em}{-}\frac{11}{3}≤ x ≤ 1$$ $$\left[-\frac{11}{3}, 1\right]$$ 
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#4:
Solutions:
$$a)\hspace{.2em}{-}5 ≤ x ≤ \frac{22}{5}$$ $$\left[-5, \frac{22}{5}\right]$$ 
$$b)\hspace{.2em}\text{All Real Numbers}$$ $$(-\infty, \infty)$$ 
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#5:
Solutions:
$$a)\hspace{.2em}x ≤ -\frac{26}{7}\hspace{.2em}\text{or} \hspace{.2em}x ≥ 6$$ $$\left(-\infty, -\frac{26}{7}\right] ∪ [6, \infty)$$ 
$$b)\hspace{.2em}\text{No Solution} \\, ∅$$
