About Solving Absolute Value Inequalities:
Recall that the absolute value of a number is the distance from that number to zero on the number line. When we see an absolute value inequality such as |x| > 5, we find our solution as all the numbers whose absolute value is greater than 5. In other words, we're looking for numbers that are more than 5 units away from zero in either direction. The answer would be x < -5 or x > 5. Similarly, if we look at |x| < 5, we find our solution as all the numbers whose absolute value is less than 5. In other words, we're looking for numbers that are less than 5 units away from zero on the number line. The answer would be -5 < x < 5.
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:
Instructions: Solve each inequality, write the solution in interval notation, and graph the interval.
a) 2|5 - 2a| + 6 < 4
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#5:
Instructions: Solve each inequality, write the solution in interval notation, and graph the interval.
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) $$\frac{1}{2}≤ v ≤ \frac{5}{2}$$
$$\left[\frac{1}{2},\frac{5}{2}\right]$$
