Solve the algebraic equations involving fractions and variables.

Let's solve each equation one by one. We'll go through the process step-by-step for clarity.

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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 both sides by 6:

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

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

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

$$
5x + 14 = 24
$$

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

✔ Answer: $x = 2$

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

LCD of 6 and 8 is 24.

Multiply both sides by 24:

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

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

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

$$
x + 4 = 216 \Rightarrow x = 212
$$

✔ Answer: $x = 212$

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

LCD of 2 and 8 is 8.

Multiply both sides by 8:

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

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

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

$$
5x + 12 = 32 \Rightarrow 5x = 20 \Rightarrow x = 4
$$

✔ Answer: $x = 4$

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

Simplify numerators first:

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

LCD of 3 and 4 is 12.

Multiply both sides by 12:

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

$$
4(x-1) - 3(x-2) = 108
$$

$$
4x - 4 - 3x + 6 = 108
$$

$$
x + 2 = 108 \Rightarrow x = 106
$$

✔ Answer: $x = 106$

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

Same as equation 3, but RHS is 9 instead of 4.

Use same method:

LCD = 8

Multiply both sides by 8:

$$
8 \left( \frac{x+2}{2} + \frac{x+4}{8} \right) = 72
\Rightarrow 4(x+2) + (x+4) = 72
$$

$$
4x + 8 + x + 4 = 72
\Rightarrow 5x + 12 = 72 \Rightarrow 5x = 60 \Rightarrow x = 12
$$

✔ Answer: $x = 12$

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

Simplify:

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

LCD of 6 and 8 is 24.

Multiply both sides by 24:

$$
24 \left( \frac{4x-2}{6} - \frac{4x-4}{8} \right) = 24 \cdot 9
\Rightarrow 4(4x-2) - 3(4x-4) = 216
$$

$$
16x - 8 - 12x + 12 = 216
\Rightarrow 4x + 4 = 216 \Rightarrow 4x = 212 \Rightarrow x = 53
$$

✔ Answer: $x = 53$

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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:

$$
24 \left( \frac{x+2}{6} + \frac{x+4}{8} \right) = 216
\Rightarrow 4(x+2) + 3(x+4) = 216
$$

$$
4x + 8 + 3x + 12 = 216
\Rightarrow 7x + 20 = 216 \Rightarrow 7x = 196 \Rightarrow x = 28
$$

✔ Answer: $x = 28$

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

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 both sides by 6:

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

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

✔ Answer: $x = -55$

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

LCD = 24

Multiply both sides by 24:

$$
24 \left( \frac{x+2}{6} - \frac{x+4}{8} \right) = 216
\Rightarrow 4(x+2) - 3(x+4) = 216
$$

$$
4x + 8 - 3x - 12 = 216 \Rightarrow x - 4 = 216 \Rightarrow x = 220
$$

✔ Answer: $x = 220$

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10. $\frac{4x-4}{8} - \frac{4x-2}{6} = 1$

We already simplified this in #8:

$$
\frac{x-1}{2} - \frac{4x-2}{6} = 1
$$

LCD = 6

Multiply by 6:

$$
6 \left( \frac{x-1}{2} - \frac{4x-2}{6} \right) = 6
\Rightarrow 3(x-1) - (4x-2) = 6
\Rightarrow 3x - 3 - 4x + 2 = 6
\Rightarrow -x -1 = 6 \Rightarrow -x = 7 \Rightarrow x = -7
$$

✔ Answer: $x = -7$

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

LCD = 24

Multiply both sides by 24:

$$
24 \left( \frac{x+2}{6} - \frac{x-4}{8} \right) = 216
\Rightarrow 4(x+2) - 3(x-4) = 216
$$

$$
4x + 8 - 3x + 12 = 216 \Rightarrow x + 20 = 216 \Rightarrow x = 196
$$

✔ Answer: $x = 196$

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12. $\frac{4x-4}{8} - \frac{4x-2}{6} = \frac{1}{2}$

This is similar to previous ones.

Simplify:

$$
\frac{x-1}{2} - \frac{4x-2}{6} = \frac{1}{2}
$$

LCD = 6

Multiply both sides by 6:

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

✔ Answer: $x = -4$

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✔ Final Answers Summary:

| Equation | Solution |
|--------|---------|
| 1. $\frac{x+2}{2} + \frac{x+4}{3} = 4$ | $x = 2$ |
| 2. $\frac{x-2}{6} - \frac{x-4}{8} = 9$ | $x = 212$ |
| 3. $\frac{x+2}{2} + \frac{x+4}{8} = 4$ | $x = 4$ |
| 4. $\frac{2x-2}{6} - \frac{2x-4}{8} = 9$ | $x = 106$ |
| 5. $\frac{x+2}{2} + \frac{x+4}{8} = 9$ | $x = 12$ |
| 6. $\frac{4x-2}{6} - \frac{4x-4}{8} = 9$ | $x = 53$ |
| 7. $\frac{x+2}{6} + \frac{x+4}{8} = 9$ | $x = 28$ |
| 8. $\frac{4x-4}{8} - \frac{4x-2}{6} = 9$ | $x = -55$ |
| 9. $\frac{x+2}{6} - \frac{x+4}{8} = 9$ | $x = 220$ |
| 10. $\frac{4x-4}{8} - \frac{4x-2}{6} = 1$ | $x = -7$ |
| 11. $\frac{x+2}{6} - \frac{x-4}{8} = 9$ | $x = 196$ |
| 12. $\frac{4x-4}{8} - \frac{4x-2}{6} = \frac{1}{2}$ | $x = -4$ |

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