Worksheet with ten equations involving fractions to solve.

Let's solve each of these equations step by step. All involve fractions and require finding a common denominator, eliminating denominators, and solving for $ x $. We'll go through them one at a time.

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1. $\frac{x+2}{2} + \frac{x+4}{3} = 4$

Step 1: Find the least common denominator (LCD) of 2 and 3 → LCD = 6

Multiply every term by 6 to eliminate denominators:

$$
6 \cdot \left( \frac{x+2}{2} + \frac{x+4}{3} \right) = 6 \cdot 4
$$

$$
3(x+2) + 2(x+4) = 24
$$

Step 2: Expand:

$$
3x + 6 + 2x + 8 = 24
$$

$$
5x + 14 = 24
$$

Step 3: Solve:

$$
5x = 10 \Rightarrow x = 2
$$

✔ Answer: $ x = 2 $

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2. $\frac{x+2}{2} + \frac{x+4}{8} = 4$

LCD of 2 and 8 is 8.

Multiply every term by 8:

$$
8 \cdot \left( \frac{x+2}{2} + \frac{x+4}{8} \right) = 8 \cdot 4
$$

$$
4(x+2) + (x+4) = 32
$$

Expand:

$$
4x + 8 + x + 4 = 32
\Rightarrow 5x + 12 = 32
$$

$$
5x = 20 \Rightarrow x = 4
$$

✔ Answer: $ x = 4 $

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3. $\frac{x+2}{2} + \frac{x+4}{8} = 9$

Same as #2, but RHS is 9.

Multiply by 8:

$$
4(x+2) + (x+4) = 72
\Rightarrow 4x + 8 + x + 4 = 72
\Rightarrow 5x + 12 = 72
$$

$$
5x = 60 \Rightarrow x = 12
$$

✔ Answer: $ x = 12 $

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4. $\frac{x+2}{6} + \frac{x+4}{8} = 9$

LCD of 6 and 8 is 24.

Multiply every term by 24:

$$
24 \cdot \left( \frac{x+2}{6} + \frac{x+4}{8} \right) = 24 \cdot 9
$$

$$
4(x+2) + 3(x+4) = 216
$$

Expand:

$$
4x + 8 + 3x + 12 = 216
\Rightarrow 7x + 20 = 216
$$

$$
7x = 196 \Rightarrow x = 28
$$

✔ Answer: $ x = 28 $

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7. $\frac{x-2}{6} - \frac{x-4}{8} = 9$

LCD of 6 and 8 is 24.

Multiply every term by 24:

$$
24 \cdot \left( \frac{x-2}{6} - \frac{x-4}{8} \right) = 24 \cdot 9
$$

$$
4(x-2) - 3(x-4) = 216
$$

Expand:

$$
4x - 8 - 3x + 12 = 216
\Rightarrow x + 4 = 216
\Rightarrow x = 212
$$

✔ Answer: $ x = 212 $

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8. $\frac{2x-2}{6} - \frac{2x-4}{8} = 9$

Simplify numerators:

$$
\frac{2(x-1)}{6} - \frac{2(x-2)}{8} = 9
\Rightarrow \frac{x-1}{3} - \frac{x-2}{4} = 9
$$

LCD of 3 and 4 is 12.

Multiply every term by 12:

$$
12 \cdot \left( \frac{x-1}{3} - \frac{x-2}{4} \right) = 12 \cdot 9
\Rightarrow 4(x-1) - 3(x-2) = 108
$$

Expand:

$$
4x - 4 - 3x + 6 = 108
\Rightarrow x + 2 = 108
\Rightarrow x = 106
$$

✔ Answer: $ x = 106 $

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9. $\frac{4x-2}{6} - \frac{4x-4}{8} = 9$

Simplify:

$$
\frac{2(2x-1)}{6} - \frac{4(x-1)}{8} = 9
\Rightarrow \frac{2x-1}{3} - \frac{x-1}{2} = 9
$$

LCD of 3 and 2 is 6.

Multiply every term by 6:

$$
6 \cdot \left( \frac{2x-1}{3} - \frac{x-1}{2} \right) = 6 \cdot 9
\Rightarrow 2(2x-1) - 3(x-1) = 54
$$

Expand:

$$
4x - 2 - 3x + 3 = 54
\Rightarrow x + 1 = 54
\Rightarrow x = 53
$$

✔ Answer: $ x = 53 $

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10. $\frac{4x-4}{8} - \frac{4x-2}{6} = 9$

Simplify:

$$
\frac{4(x-1)}{8} - \frac{4x-2}{6} = 9
\Rightarrow \frac{x-1}{2} - \frac{4x-2}{6} = 9
$$

LCD of 2 and 6 is 6.

Multiply every term by 6:

$$
6 \cdot \left( \frac{x-1}{2} - \frac{4x-2}{6} \right) = 6 \cdot 9
\Rightarrow 3(x-1) - (4x-2) = 54
$$

Expand:

$$
3x - 3 - 4x + 2 = 54
\Rightarrow -x -1 = 54
\Rightarrow -x = 55
\Rightarrow x = -55
$$

✔ Answer: $ x = -55 $

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✔ Final Answers:

1. $ x = 2 $
2. $ x = 4 $
3. $ x = 12 $
4. $ x = 28 $
7. $ x = 212 $
8. $ x = 106 $
9. $ x = 53 $
10. $ x = -55 $

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