### 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

#1:

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

a) 5 - 3|9p + 9| ≤ -103

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#2:

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

a) $$\frac{3}{2}\left|1 + \frac{1}{3}x\right| + \frac{5}{2}≥ \frac{8}{3}$$

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#3:

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

a) 9 + 4|8n - 2| ≤ 49

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#4:

a) 2|5 - 2a| + 6 < 4

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#5:

a) $$-\frac{3}{2}\left| -1 + \frac{2}{3}v \right| + 2 ≥ 1$$

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Written Solutions:

#1:

Solutions:

a) p ≤ -5 or p ≥ 3

(-∞,-5] ∪ [3,∞)

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#2:

Solutions:

a) $$x ≤ -\frac{10}{3}$$ or $$x ≥ -\frac{8}{3}$$

$$\left(-∞,-\frac{10}{3}\right] ∪ \left[-\frac{8}{3}, ∞\right)$$

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#3:

Solutions:

a) $$-1 ≤ n ≤ \frac{3}{2}$$

$$\left[-1,\frac{3}{2}\right]$$

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#4:

Solutions:

a) No solution: ∅

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#5:

Solutions:

a) $$\frac{1}{2}≤ v ≤ \frac{5}{2}$$

$$\left[\frac{1}{2},\frac{5}{2}\right]$$