Question

1. $|5x| + 5 = 45$
2. $3|-8x| + 8 = 80$
3. $5 - 8|-2n| = -75$
4. $-5|3 + 4k| = -115$
5. $\frac{|7p + 4|}{8} = 3$
6. $3 - |8x - 6| = 3$
7. $2 - 5|5m - 5| = -73$
8. $6|1 - 5x| - 9 = 57$
9. $3|3 - 5r| - 3 = 18$
10. $5|9 - 5n| - 7 = 38$

Explanation:

Problem 11: $|5x| + 5 = 45$

Step1: Isolate the absolute value

$|5x| = 45 - 5$
$|5x| = 40$

Step2: Split into two equations

$5x = 40$ or $5x = -40$

Step3: Solve for $x$

$x = \frac{40}{5} = 8$ or $x = \frac{-40}{5} = -8$

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Problem 12: $3|-8x| + 8 = 80$

Step1: Isolate the absolute value

$3|-8x| = 80 - 8$
$3|-8x| = 72$
$|-8x| = \frac{72}{3} = 24$

Step2: Simplify absolute value

$|8x| = 24$

Step3: Split into two equations

$8x = 24$ or $8x = -24$

Step4: Solve for $x$

$x = \frac{24}{8} = 3$ or $x = \frac{-24}{8} = -3$

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Problem 13: $5 - 8|-2n| = -75$

Step1: Isolate the absolute value

$-8|-2n| = -75 - 5$
$-8|-2n| = -80$
$|-2n| = \frac{-80}{-8} = 10$

Step2: Simplify absolute value

$|2n| = 10$

Step3: Split into two equations

$2n = 10$ or $2n = -10$

Step4: Solve for $n$

$n = \frac{10}{2} = 5$ or $n = \frac{-10}{2} = -5$

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Problem 14: $-5|3 + 4k| = -115$

Step1: Isolate the absolute value

$|3 + 4k| = \frac{-115}{-5} = 23$

Step2: Split into two equations

$3 + 4k = 23$ or $3 + 4k = -23$

Step3: Solve for $k$

$4k = 23 - 3 = 20 \implies k = 5$
$4k = -23 - 3 = -26 \implies k = \frac{-26}{4} = -\frac{13}{2}$

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Problem 15: $\frac{|7p + 4|}{8} = 3$

Step1: Eliminate denominator

$|7p + 4| = 3 \times 8 = 24$

Step2: Split into two equations

$7p + 4 = 24$ or $7p + 4 = -24$

Step3: Solve for $p$

$7p = 24 - 4 = 20 \implies p = \frac{20}{7}$
$7p = -24 - 4 = -28 \implies p = \frac{-28}{7} = -4$

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Problem 16: $3 - |8x - 6| = 3$

Step1: Isolate the absolute value

$-|8x - 6| = 3 - 3 = 0$
$|8x - 6| = 0$

Step2: Solve the single equation

$8x - 6 = 0$
$8x = 6 \implies x = \frac{6}{8} = \frac{3}{4}$

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Problem 17: $2 - 5|5m - 5| = -73$

Step1: Isolate the absolute value

$-5|5m - 5| = -73 - 2 = -75$
$|5m - 5| = \frac{-75}{-5} = 15$

Step2: Split into two equations

$5m - 5 = 15$ or $5m - 5 = -15$

Step3: Solve for $m$

$5m = 15 + 5 = 20 \implies m = 4$
$5m = -15 + 5 = -10 \implies m = -2$

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Problem 18: $6|1 - 5x| - 9 = 57$

Step1: Isolate the absolute value

$6|1 - 5x| = 57 + 9 = 66$
$|1 - 5x| = \frac{66}{6} = 11$

Step2: Split into two equations

$1 - 5x = 11$ or $1 - 5x = -11$

Step3: Solve for $x$

$-5x = 11 - 1 = 10 \implies x = -2$
$-5x = -11 - 1 = -12 \implies x = \frac{12}{5}$

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Problem 19: $3|3 - 5r| - 3 = 18$

Step1: Isolate the absolute value

$3|3 - 5r| = 18 + 3 = 21$
$|3 - 5r| = \frac{21}{3} = 7$

Step2: Split into two equations

$3 - 5r = 7$ or $3 - 5r = -7$

Step3: Solve for $r$

$-5r = 7 - 3 = 4 \implies r = -\frac{4}{5}$
$-5r = -7 - 3 = -10 \implies r = 2$

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Problem 20: $5|9 - 5n| - 7 = 38$

Step1: Isolate the absolute value

$5|9 - 5n| = 38 + 7 = 45$
$|9 - 5n| = \frac{45}{5} = 9$

Step2: Split into two equations

$9 - 5n = 9$ or $9 - 5n = -9$

Step3: Solve for $n$

$-5n = 9 - 9 = 0 \implies n = 0$
$-5n = -9 - 9 = -18 \implies n = \frac{18}{5}$

Answer:

1. $x = 8$ or $x = -8$
2. $x = 3$ or $x = -3$
3. $n = 5$ or $n = -5$
4. $k = 5$ or $k = -\frac{13}{2}$
5. $p = \frac{20}{7}$ or $p = -4$
6. $x = \frac{3}{4}$
7. $m = 4$ or $m = -2$
8. $x = -2$ or $x = \frac{12}{5}$
9. $r = -\frac{4}{5}$ or $r = 2$
10. $n = 0$ or $n = \frac{18}{5}$