Question

algebra 2
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absolute value equations and inequalities cw 1
name ______
date ______
solve each equation. check for extraneous solutions.

1. |10m - 5| = 5
2. 8 - 5|-6k| = 38
3. |-4b - 9| - 8 = 41
4. 4 + 9|5v + 4| = 103
5. \\(\frac{2|6x - 24|}{9} + 5 = 17\\)
6. 5n - 3 = |n + 9|
7. |x + 5| = 2x - 1
8. 4 - 2n = |2 + 3n|

Explanation:

1) Equation: $|10m - 5| = 5$

Step1: Split into two cases

Case 1: $10m - 5 = 5$
Case 2: $10m - 5 = -5$

Step2: Solve Case 1

$10m = 5 + 5 = 10$
$m = \frac{10}{10} = 1$

Step3: Solve Case 2

$10m = -5 + 5 = 0$
$m = \frac{0}{10} = 0$

Step4: Verify solutions

For $m=1$: $|10(1)-5|=|5|=5$ (valid)
For $m=0$: $|10(0)-5|=|-5|=5$ (valid)

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2) Equation: $8 - 5|-6k| = 38$

Step1: Isolate absolute value term

$-5|-6k| = 38 - 8 = 30$
$|-6k| = \frac{30}{-5} = -6$

Step2: Analyze validity

Absolute value cannot be negative, so no solution.

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3) Equation: $|-4b - 9| - 8 = 41$

Step1: Isolate absolute value term

$|-4b - 9| = 41 + 8 = 49$

Step2: Split into two cases

Case1: $-4b - 9 = 49$
Case2: $-4b - 9 = -49$

Step3: Solve Case1

$-4b = 49 + 9 = 58$
$b = \frac{58}{-4} = -\frac{29}{2}$

Step4: Solve Case2

$-4b = -49 + 9 = -40$
$b = \frac{-40}{-4} = 10$

Step5: Verify solutions

For $b=-\frac{29}{2}$: $|-4(-\frac{29}{2})-9|=|58-9|=|49|=49$ (valid)
For $b=10$: $|-4(10)-9|=|-49|=49$ (valid)

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4) Equation: $4 + 9|5v + 4| = 103$

Step1: Isolate absolute value term

$9|5v + 4| = 103 - 4 = 99$
$|5v + 4| = \frac{99}{9} = 11$

Step2: Split into two cases

Case1: $5v + 4 = 11$
Case2: $5v + 4 = -11$

Step3: Solve Case1

$5v = 11 - 4 = 7$
$v = \frac{7}{5}$

Step4: Solve Case2

$5v = -11 - 4 = -15$
$v = \frac{-15}{5} = -3$

Step5: Verify solutions

For $v=\frac{7}{5}$: $|5(\frac{7}{5})+4|=|7+4|=11$ (valid)
For $v=-3$: $|5(-3)+4|=|-15+4|=|-11|=11$ (valid)

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5) Equation: $\frac{2|6x - 24|}{9} + 5 = 17$

Step1: Isolate absolute value term

$\frac{2|6x - 24|}{9} = 17 - 5 = 12$
$2|6x - 24| = 12 \times 9 = 108$
$|6x - 24| = \frac{108}{2} = 54$

Step2: Split into two cases

Case1: $6x - 24 = 54$
Case2: $6x - 24 = -54$

Step3: Solve Case1

$6x = 54 + 24 = 78$
$x = \frac{78}{6} = 13$

Step4: Solve Case2

$6x = -54 + 24 = -30$
$x = \frac{-30}{6} = -5$

Step5: Verify solutions

For $x=13$: $|6(13)-24|=|78-24|=54$ (valid)
For $x=-5$: $|6(-5)-24|=|-30-24|=|-54|=54$ (valid)

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6) Equation: $5n - 3 = |n + 9|$

Step1: Split into two cases

Case1: $n + 9 \geq 0$ (i.e., $n \geq -9$): $5n - 3 = n + 9$
Case2: $n + 9 < 0$ (i.e., $n < -9$): $5n - 3 = -(n + 9)$

Step2: Solve Case1

$5n - n = 9 + 3$
$4n = 12$
$n = 3$ (satisfies $n \geq -9$)

Step3: Solve Case2

$5n - 3 = -n - 9$
$5n + n = -9 + 3$
$6n = -6$
$n = -1$ (does not satisfy $n < -9$, extraneous)

Step4: Verify valid solution

For $n=3$: $5(3)-3=12$, $|3+9|=12$ (valid)

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7) Equation: $|x + 5| = 2x - 1$

Step1: Note right-hand side constraint

$2x - 1 \geq 0 \implies x \geq \frac{1}{2}$

Step2: Split into two cases

Case1: $x + 5 \geq 0$ (i.e., $x \geq -5$): $x + 5 = 2x - 1$
Case2: $x + 5 < 0$ (i.e., $x < -5$): $x + 5 = -(2x - 1)$

Step3: Solve Case1

$5 + 1 = 2x - x$
$x = 6$ (satisfies $x \geq \frac{1}{2}$)

Step4: Solve Case2

$x + 5 = -2x + 1$
$x + 2x = 1 - 5$
$3x = -4$
$x = -\frac{4}{3}$ (does not satisfy $x \geq \frac{1}{2}$, extraneous)

Step5: Verify valid solution

For $x=6$: $|6+5|=11$, $2(6)-1=11$ (valid)

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8) Equation: $4 - 2n = |2 + 3n|$

Step1: Note right-hand side constraint

$4 - 2n \geq 0 \implies n \leq 2$

Step2: Split into two cases

Case1: $2 + 3n \geq 0$ (i.e., $n \geq -\frac{2}{3}$): $4 - 2n = 2 + 3n$
Case2: $2 + 3n < 0$ (i.e., $n < -\frac{2}{3}$): $4 - 2n = -(2 + 3n)$

Step3: Solve Case1

$4 - 2 = 3n + 2n$
$2 = 5n$
$n = \frac{2}{5}$ (satisfies $n \leq 2$)

Step4: Solve Case2

$4 - 2n = -2 - 3n$
$-2n + 3n = -2 - 4$
$n = -6$ (satisfies $n \leq 2$)

Step5: Verify solutions

For $n=\frac{2}{5}$: $4-2(\frac{2}{5})=\frac{16}{5}$, $|2+3(\frac{2}{5})|=|\frac{16}{5}|=\frac{16}{5}$ (valid)
For $n=-6$: $4-2(-6)=16$, $|2+3(-6)|=|-16|=16$ (valid)

Answer:

1. $m=0$ and $m=1$
2. No solution
3. $b=-\frac{29}{2}$ and $b=10$
4. $v=\frac{7}{5}$ and $v=-3$
5. $x=13$ and $x=-5$
6. $n=3$
7. $x=6$
8. $n=\frac{2}{5}$ and $n=-6$