Worksheet with ten equations involving fractions to solve.

Let's solve each of these equations step by step. These are all linear equations involving fractions. The key strategy is to eliminate the denominators by multiplying both sides of the equation by the least common denominator (LCD), then simplify and solve for $ x $.

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

Step 1: Find LCD of 2 and 3 → 6

Multiply both sides by 6:

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

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

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 both sides by 8:

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

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

$$
4x + 8 + x + 4 = 32
$$

$$
5x + 12 = 32
$$

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

✔ 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:

$$
4(x+2) + (x+4) = 72
$$

$$
4x + 8 + x + 4 = 72
$$

$$
5x + 12 = 72
$$

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

✔ 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 both sides by 24:

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

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

$$
4x + 8 + 3x + 12 = 216
$$

$$
7x + 20 = 216
$$

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

✔ 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 both sides by 24:

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

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

$$
4x - 8 - 3x + 12 = 216
$$

$$
x + 4 = 216
$$

$$
x = 212
$$

✔ Answer: $ x = 212 $

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

Simplify numerator expressions first (optional):

Factor:
- $2x - 2 = 2(x - 1)$
- $2x - 4 = 2(x - 2)$

But we can just use LCD = 24

Multiply both sides by 24:

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

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

$$
8x - 8 - 6x + 12 = 216
$$

$$
2x + 4 = 216
$$

$$
2x = 212 \Rightarrow x = 106
$$

✔ Answer: $ x = 106 $

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

LCD of 6 and 8 is 24

Multiply both sides by 24:

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

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

$$
16x - 8 - 12x + 12 = 216
$$

$$
4x + 4 = 216
$$

$$
4x = 212 \Rightarrow x = 53
$$

✔ Answer: $ x = 53 $

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

Note: This is similar to #9, but signs are reversed.

LCD = 24

Multiply both sides by 24:

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

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

$$
12x - 12 - 16x + 8 = 216
$$

$$
-4x - 4 = 216
$$

$$
-4x = 220 \Rightarrow x = -55
$$

✔ Answer: $ x = -55 $

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✔ Final Answers Summary:

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

Let me know if you'd like a visual explanation or check for any of them!