Question

1. solve for variable

a) $5|9 - 5n| - 7 = 38$
b) $\frac{1}{3}|k + 2| + 4 = \frac{1}{3}$
c) $3|2x - 7| + 10 = 10$

1. solve for the variable

a) $4|6 - 2a| + 8 \leq 24$
b) $9|3n - 2| + 6 > 51$
c) $9|r - 2| - 10 < -73$

Explanation:

15a Step1: Isolate the absolute value term

Add 7 to both sides:
$5|9-5n| = 38 + 7 = 45$

15a Step2: Simplify the equation

Divide both sides by 5:
$|9-5n| = \frac{45}{5} = 9$

15a Step3: Split into two cases

Case 1: $9-5n = 9$
Case 2: $9-5n = -9$

15a Step4: Solve Case 1

Subtract 9, divide by -5:
$-5n = 9 - 9 = 0 \implies n = 0$

15a Step5: Solve Case 2

Subtract 9, divide by -5:
$-5n = -9 - 9 = -18 \implies n = \frac{-18}{-5} = \frac{18}{5}$

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15b Step1: Isolate the absolute value term

Subtract 4 from both sides:
$\frac{1}{3}|k+2| = \frac{1}{3} - 4 = \frac{1}{3} - \frac{12}{3} = -\frac{11}{3}$

15b Step2: Multiply by 3

$|k+2| = -11$

15b Step3: Analyze absolute value

Absolute value cannot be negative, so no solution.

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15c Step1: Isolate the absolute value term

Subtract 10 from both sides:
$3|2x-7| = 10 - 10 = 0$

15c Step2: Simplify the equation

Divide by 3:
$|2x-7| = 0$

15c Step3: Solve for x

Set inside equal to 0:
$2x-7 = 0 \implies 2x = 7 \implies x = \frac{7}{2}$

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16a Step1: Isolate the absolute value term

Subtract 8 from both sides:
$4|6-2a| \leq 24 - 8 = 16$

16a Step2: Simplify the inequality

Divide by 4:
$|6-2a| \leq 4$

16a Step3: Rewrite as compound inequality

$-4 \leq 6-2a \leq 4$

16a Step4: Subtract 6 from all parts

$-4 - 6 \leq -2a \leq 4 - 6 \implies -10 \leq -2a \leq -2$

16a Step5: Divide by -2 (reverse inequalities)

$5 \geq a \geq 1$ or $1 \leq a \leq 5$

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16b Step1: Isolate the absolute value term

Subtract 6 from both sides:
$9|3n-2| > 51 - 6 = 45$

16b Step2: Simplify the inequality

Divide by 9:
$|3n-2| > 5$

16b Step3: Split into two inequalities

Case 1: $3n-2 > 5$
Case 2: $3n-2 < -5$

16b Step4: Solve Case 1

Add 2, divide by 3:
$3n > 7 \implies n > \frac{7}{3}$

16b Step5: Solve Case 2

Add 2, divide by 3:
$3n < -3 \implies n < -1$

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16c Step1: Isolate the absolute value term

Add 10 to both sides:
$9|r-2| < -73 + 10 = -63$

16c Step2: Simplify the inequality

Divide by 9:
$|r-2| < -7$

16c Step3: Analyze absolute value

Absolute value cannot be negative, so no solution.

Answer:

15a) $n = 0$ or $n = \frac{18}{5}$
15b) No solution
15c) $x = \frac{7}{2}$
16a) $1 \leq a \leq 5$
16b) $n < -1$ or $n > \frac{7}{3}$
16c) No solution