Worksheet for solving equations and inequalities, including linear equations and basic algebraic expressions.

Here is the complete, step-by-step solution to Chapter 6 Unit Exam: Equations and Inequalities.

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1. Solve the following. Show all steps clearly.

a) \( x + 5 = -4 \)
Subtract 5 from both sides:
\( x + 5 - 5 = -4 - 5 \)
→ \( x = -9 \)

✔ Answer: \( x = -9 \)

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b) \( r - 3 = 4 \)
Add 3 to both sides:
\( r - 3 + 3 = 4 + 3 \)
→ \( r = 7 \)

✔ Answer: \( r = 7 \)

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c) \( -3 + j = 21 \)
Add 3 to both sides:
\( -3 + j + 3 = 21 + 3 \)
→ \( j = 24 \)

✔ Answer: \( j = 24 \)

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d) \( 3 = n + 5 \)
Subtract 5 from both sides:
\( 3 - 5 = n + 5 - 5 \)
→ \( -2 = n \) or \( n = -2 \)

✔ Answer: \( n = -2 \)

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2. Solve the following. Show all steps clearly.

a) \( 2j = 6 \)
Divide both sides by 2:
\( \frac{2j}{2} = \frac{6}{2} \)
→ \( j = 3 \)

✔ Answer: \( j = 3 \)

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b) \( 25x = -50 \)
Divide both sides by 25:
\( \frac{25x}{25} = \frac{-50}{25} \)
→ \( x = -2 \)

✔ Answer: \( x = -2 \)

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c) \( 3r = 27 \)
Divide both sides by 3:
\( \frac{3r}{3} = \frac{27}{3} \)
→ \( r = 9 \)

✔ Answer: \( r = 9 \)

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d) \( 44 = -4n \)
Divide both sides by -4:
\( \frac{44}{-4} = \frac{-4n}{-4} \)
→ \( -11 = n \) or \( n = -11 \)

✔ Answer: \( n = -11 \)

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3. Solve the following. Show all steps clearly.

a) \( 8 = \frac{x}{4} \)
Multiply both sides by 4:
\( 8 \cdot 4 = \frac{x}{4} \cdot 4 \)
→ \( 32 = x \) or \( x = 32 \)

✔ Answer: \( x = 32 \)

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b) \( \frac{x}{2} = 4 \)
Multiply both sides by 2:
\( \frac{x}{2} \cdot 2 = 4 \cdot 2 \)
→ \( x = 8 \)

✔ Answer: \( x = 8 \)

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c) \( \frac{x}{3} = 6 \)
Multiply both sides by 3:
\( \frac{x}{3} \cdot 3 = 6 \cdot 3 \)
→ \( x = 18 \)

✔ Answer: \( x = 18 \)

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d) \( \frac{x}{2} = 5 \)
Multiply both sides by 2:
\( \frac{x}{2} \cdot 2 = 5 \cdot 2 \)
→ \( x = 10 \)

✔ Answer: \( x = 10 \)

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4. Solve the following. Show all steps clearly.

a) \( 2x = 2 + 4x \)
Subtract 4x from both sides:
\( 2x - 4x = 2 + 4x - 4x \)
→ \( -2x = 2 \)
Divide both sides by -2:
\( x = \frac{2}{-2} = -1 \)

✔ Answer: \( x = -1 \)

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b) \( 5x + 2x = 35 \)
Combine like terms:
\( 7x = 35 \)
Divide both sides by 7:
\( x = \frac{35}{7} = 5 \)

✔ Answer: \( x = 5 \)

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c) \( -3 + j = -6j + 18 \)
Add 6j to both sides:
\( -3 + j + 6j = -6j + 18 + 6j \)
→ \( -3 + 7j = 18 \)
Add 3 to both sides:
\( 7j = 21 \)
Divide by 7:
\( j = 3 \)

✔ Answer: \( j = 3 \)

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d) \( 6n - 10 = n + 5 \)
Subtract n from both sides:
\( 6n - n - 10 = n - n + 5 \)
→ \( 5n - 10 = 5 \)
Add 10 to both sides:
\( 5n = 15 \)
Divide by 5:
\( n = 3 \)

✔ Answer: \( n = 3 \)

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5. Solve the following. Show all steps clearly.

a) \( 2(x + 1) = -4 \)
Distribute the 2:
\( 2x + 2 = -4 \)
Subtract 2 from both sides:
\( 2x = -6 \)
Divide by 2:
\( x = -3 \)

✔ Answer: \( x = -3 \)

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b) \( 2(x + 4) = 14 \)
Distribute the 2:
\( 2x + 8 = 14 \)
Subtract 8 from both sides:
\( 2x = 6 \)
Divide by 2:
\( x = 3 \)

✔ Answer: \( x = 3 \)

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c) \( 2(4x - 5) = 2x + 2 \)
Distribute the 2 on the left:
\( 8x - 10 = 2x + 2 \)
Subtract 2x from both sides:
\( 6x - 10 = 2 \)
Add 10 to both sides:
\( 6x = 12 \)
Divide by 6:
\( x = 2 \)

✔ Answer: \( x = 2 \)

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d) \( 12 + x = 3(x + 2) \)
Distribute the 3 on the right:
\( 12 + x = 3x + 6 \)
Subtract x from both sides:
\( 12 = 2x + 6 \)
Subtract 6 from both sides:
\( 6 = 2x \)
Divide by 2:
\( x = 3 \)

✔ Answer: \( x = 3 \)

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## ✔ Final Answers Summary:

1.

a) \( x = -9 \)
b) \( r = 7 \)
c) \( j = 24 \)
d) \( n = -2 \)

2.

a) \( j = 3 \)
b) \( x = -2 \)
c) \( r = 9 \)
d) \( n = -11 \)

3.

a) \( x = 32 \)
b) \( x = 8 \)
c) \( x = 18 \)
d) \( x = 10 \)

4.

a) \( x = -1 \)
b) \( x = 5 \)
c) \( j = 3 \)
d) \( n = 3 \)

5.

a) \( x = -3 \)
b) \( x = 3 \)
c) \( x = 2 \)
d) \( x = 3 \)

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