Question: Add the following:
1) $\frac{4}{9}x + \frac{1}{6}y$
2) $\frac{3}{x + 2} + \frac{5}{x - 4}$
3) $\frac{2}{x + 1} + \frac{6}{x + 3}$
4) $\frac{7}{x^2 - 25} + \frac{3}{x - 5}$
5) $\frac{4}{x^2 - 4} + \frac{1}{x + 2}$
### Problem 3: Add $\frac{2}{x + 1} + \frac{6}{x + 3}$
1. **State the problem:** Add the two rational expressions $\frac{2}{x + 1}$ and $\frac{6}{x + 3}$.
2. **Formula and rules:** To add fractions, find a common denominator, then add the numerators.
3. **Find the common denominator:** The denominators are $x + 1$ and $x + 3$. The common denominator is their product: $$ (x + 1)(x + 3) $$
4. **Rewrite each fraction with the common denominator:**
$$ \frac{2}{x + 1} = \frac{2(x + 3)}{(x + 1)(x + 3)} $$
$$ \frac{6}{x + 3} = \frac{6(x + 1)}{(x + 1)(x + 3)} $$
5. **Add the numerators:**
$$ \frac{2(x + 3) + 6(x + 1)}{(x + 1)(x + 3)} $$
6. **Simplify the numerator:**
$$ 2x + 6 + 6x + 6 = 8x + 12 $$
7. **Final expression:**
$$ \frac{8x + 12}{(x + 1)(x + 3)} $$
8. **Factor numerator if possible:**
$$ 8x + 12 = 4(2x + 3) $$
9. **Simplified answer:**
$$ \frac{4(2x + 3)}{(x + 1)(x + 3)} $$
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### Problem 4: Add $\frac{7}{x^2 - 25} + \frac{3}{x - 5}$
1. **State the problem:** Add $\frac{7}{x^2 - 25}$ and $\frac{3}{x - 5}$.
2. **Factor denominator:**
$$ x^2 - 25 = (x - 5)(x + 5) $$
3. **Common denominator:**
$$ (x - 5)(x + 5) $$
4. **Rewrite fractions:**
$$ \frac{7}{(x - 5)(x + 5)} $$ stays the same.
$$ \frac{3}{x - 5} = \frac{3(x + 5)}{(x - 5)(x + 5)} $$
5. **Add numerators:**
$$ \frac{7 + 3(x + 5)}{(x - 5)(x + 5)} $$
6. **Simplify numerator:**
$$ 7 + 3x + 15 = 3x + 22 $$
7. **Final answer:**
$$ \frac{3x + 22}{(x - 5)(x + 5)} $$
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### Problem 5: Add $\frac{4}{x^2 - 4} + \frac{1}{x + 2}$
1. **State the problem:** Add $\frac{4}{x^2 - 4}$ and $\frac{1}{x + 2}$.
2. **Factor denominator:**
$$ x^2 - 4 = (x - 2)(x + 2) $$
3. **Common denominator:**
$$ (x - 2)(x + 2) $$
4. **Rewrite fractions:**
$$ \frac{4}{(x - 2)(x + 2)} $$ stays the same.
$$ \frac{1}{x + 2} = \frac{1(x - 2)}{(x + 2)(x - 2)} $$
5. **Add numerators:**
$$ \frac{4 + (x - 2)}{(x - 2)(x + 2)} $$
6. **Simplify numerator:**
$$ 4 + x - 2 = x + 2 $$
7. **Final answer:**
$$ \frac{x + 2}{(x - 2)(x + 2)} $$
8. **Simplify fraction by canceling $x + 2$:**
$$ \frac{\cancel{x + 2}}{(x - 2)\cancel{(x + 2)}} = \frac{1}{x - 2} $$
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### Final answers:
3) $$ \frac{4(2x + 3)}{(x + 1)(x + 3)} $$
4) $$ \frac{3x + 22}{(x - 5)(x + 5)} $$
5) $$ \frac{1}{x - 2} $$