Solving Equations and Inequalities w- Absolute Value worksheet ... - Free Printable

Let's solve each equation step by step and explain the reasoning.

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1) $-2(6 - 5n) = -82$

Step 1: Distribute the $-2$:
$$
-2 \cdot 6 + (-2) \cdot (-5n) = -82 \\
-12 + 10n = -82
$$

Step 2: Add 12 to both sides:
$$
10n = -70
$$

Step 3: Divide by 10:
$$
n = -7
$$

✔ Answer: D) $\{-7\}$

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2) $-2m + 4(-3m + 7) = -84$

Step 1: Distribute the 4:
$$
-2m + (-12m + 28) = -84 \\
-2m - 12m + 28 = -84 \\
-14m + 28 = -84
$$

Step 2: Subtract 28:
$$
-14m = -112
$$

Step 3: Divide by $-14$:
$$
m = 8
$$

✔ Answer: B) $\{8\}$

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3) $-3n - 25 = 7(4 - 8n)$

Step 1: Expand the right side:
$$
-3n - 25 = 28 - 56n
$$

Step 2: Add $56n$ to both sides:
$$
53n - 25 = 28
$$

Step 3: Add 25:
$$
53n = 53
$$

Step 4: Divide:
$$
n = 1
$$

✔ Answer: C) $\{1\}$

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4) $2m - 38 = 7(m - 4)$

Step 1: Expand the right:
$$
2m - 38 = 7m - 28
$$

Step 2: Subtract $2m$:
$$
-38 = 5m - 28
$$

Step 3: Add 28:
$$
-10 = 5m
$$

Step 4: Divide:
$$
m = -2
$$

✔ Answer: A) $\{-2\}$

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5) $|3 + b| = 5$

Absolute value means two cases:

Case 1: $3 + b = 5$ → $b = 2$

Case 2: $3 + b = -5$ → $b = -8$

So, solutions: $b = 2$ or $b = -8$

✔ Answer: C) $\{2, -8\}$

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6) $|-10n| = 20$

Since absolute value of $-10n$ is same as $|10n|$, so:
$$
|10n| = 20
\Rightarrow 10n = 20 \quad \text{or} \quad 10n = -20
\Rightarrow n = 2 \quad \text{or} \quad n = -2
$$

So $n = 2$ or $n = -2$

But look at choices — they are sets of numbers. Let’s check:

A) {3,1} — no
B) {-2,2} — yes!
C) {4,-4} — no
D) {8,-18} — no

✔ Answer: B) $\{-2, 2\}$

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7) $\left|\frac{r}{2}\right| - 1 = 4$

Step 1: Add 1:
$$
\left|\frac{r}{2}\right| = 5
$$

Step 2: Two cases:

- $\frac{r}{2} = 5$ → $r = 10$
- $\frac{r}{2} = -5$ → $r = -10$

✔ Answer: B) $\{10, -10\}$

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8) $|a + 3| - 10 = -1$

Step 1: Add 10:
$$
|a + 3| = 9
$$

Step 2: Two cases:

- $a + 3 = 9$ → $a = 6$
- $a + 3 = -9$ → $a = -12$

✔ Answer: A) $\{6, -12\}$

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9) $|10 - 7a| = 38$

Two cases:

Case 1: $10 - 7a = 38$
→ $-7a = 28$ → $a = -4$

Case 2: $10 - 7a = -38$
→ $-7a = -48$ → $a = \frac{48}{7}$

So $a = -4$ or $a = \frac{48}{7}$

Check options:

A) $\left\{2, -\frac{3}{5}\right\}$ — no
B) $\{11, -9\}$ — no
C) $\left\{-4, \frac{48}{7}\right\}$ — yes!
D) $\{2\}$ — no

✔ Answer: C) $\left\{-4, \frac{48}{7}\right\}$

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10) $\frac{|-4x - 8|}{8} = 1$

Step 1: Multiply both sides by 8:
$$
|-4x - 8| = 8
$$

Note: $|-4x - 8| = |-(4x + 8)| = |4x + 8|$, so:
$$
|4x + 8| = 8
$$

Now two cases:

Case 1: $4x + 8 = 8$ → $4x = 0$ → $x = 0$

Case 2: $4x + 8 = -8$ → $4x = -16$ → $x = -4$

So $x = 0$ or $x = -4$

Now check answer choices:

A) $\left\{-\frac{10}{3}\right\}$ — no
B) $\left\{\frac{13}{4}, -\frac{1}{4}\right\}$ — no
C) $\{-4, 0\}$ — yes!
D) $\left\{-\frac{10}{3}, \frac{34}{9}\right\}$ — no

✔ Answer: C) $\{-4, 0\}$

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✔ Final Answers:

1. D
2. B
3. C
4. A
5. C
6. B
7. B
8. A
9. C
10. C

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