Question

problem set 2: solving absolute value equations and inequalities

match each compound inequality to its equivalent absolute value inequality.

1
① $-15 < -3x + 7 < 15$
$|-3x + 7| < 15$
② $-3x + 7 \leq -15$ or $-3x + 7 \geq 15$
$|-3x + 7| \geq 15$
③ $-15 \leq -3x + 7 \leq 15$
$|-3x + 7| \leq 15$
④ $-3x + 7 < -15$ or $-3x + 7 > 15$
$|-3x + 7| > 15$
$|-3x + 7| \leq 15$
$|-3x + 7| > 15$
$|-3x + 7| < 15$
$|-3x + 7| \geq 15$
2
① $-2 \leq 6x - 5 \leq 2$
$|6x - 5| > 2$
② $6x - 5 < -2$ or $6x - 5 > 2$
$|6x - 5| \geq 2$
③ $6x - 5 \leq -2$ or $6x - 5 \geq 2$
$|6x - 5| < 2$
④ $-2 < 6x - 5 < 2$
$|6x - 5| \leq 2$
3
① $-11 < -x - 10 < 11$
$|-x - 10| \leq 11$
② $-x - 10 \leq -11$ or $-x - 10 \geq 11$
$|-x - 10| > 11$
③ $-11 \leq -x - 10 \leq 11$
$|-x - 10| < 11$
④ $-x - 10 < -11$ or $-x - 10 > 11$
$|-x - 10| \geq 11$

Explanation:

Step1: Recall absolute value rules

For any expression $A$ and positive number $k$:

1. $|A| < k$ is equivalent to $-k < A < k$
2. $|A| > k$ is equivalent to $A < -k$ or $A > k$
3. $|A| \leq k$ is equivalent to $-k \leq A \leq k$
4. $|A| \geq k$ is equivalent to $A \leq -k$ or $A \geq k$

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Problem 1 Matching

Step1: Match (a) to absolute value

$-15 < -3x+7 < 15$ matches $|-3x+7| < 15$

Step2: Match (b) to absolute value

$-3x+7 \leq -15$ or $-3x+7 \geq 15$ matches $|-3x+7| \geq 15$

Step3: Match (c) to absolute value

$-15 \leq -3x+7 \leq 15$ matches $|-3x+7| \leq 15$

Step4: Match (d) to absolute value

$-3x+7 < -15$ or $-3x+7 > 15$ matches $|-3x+7| > 15$

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Problem 2 Matching

Step1: Match (a) to absolute value

$-2 \leq 6x-5 \leq 2$ matches $|6x-5| \leq 2$

Step2: Match (b) to absolute value

$6x-5 < -2$ or $6x-5 > 2$ matches $|6x-5| > 2$

Step3: Match (c) to absolute value

$6x-5 \leq -2$ or $6x-5 \geq 2$ matches $|6x-5| \geq 2$

Step4: Match (d) to absolute value

$-2 < 6x-5 < 2$ matches $|6x-5| < 2$

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Problem 3 Matching

Step1: Match (a) to absolute value

$-11 < -x-10 < 11$ matches $|-x-10| < 11$

Step2: Match (b) to absolute value

$-x-10 \leq -11$ or $-x-10 \geq 11$ matches $|-x-10| \geq 11$

Step3: Match (c) to absolute value

$-11 \leq -x-10 \leq 11$ matches $|-x-10| \leq 11$

Step4: Match (d) to absolute value

$-x-10 < -11$ or $-x-10 > 11$ matches $|-x-10| > 11$

Answer:

Problem 1

• (a) $\boldsymbol{|-3x+7| < 15}$
• (b) $\boldsymbol{|-3x+7| \geq 15}$
• (c) $\boldsymbol{|-3x+7| \leq 15}$
• (d) $\boldsymbol{|-3x+7| > 15}$

Problem 2

• (a) $\boldsymbol{|6x-5| \leq 2}$
• (b) $\boldsymbol{|6x-5| > 2}$
• (c) $\boldsymbol{|6x-5| \geq 2}$
• (d) $\boldsymbol{|6x-5| < 2}$

Problem 3

• (a) $\boldsymbol{|-x-10| < 11}$
• (b) $\boldsymbol{|-x-10| \geq 11}$
• (c) $\boldsymbol{|-x-10| \leq 11}$
• (d) $\boldsymbol{|-x-10| > 11}$