Question: Add the following:
1) 4/9x + 1/6y
2) 3/(x + 2) + 5/(x - 4)
3) 2/(x + 1) + 6/(x + 3)
4) 7/(x^2 - 25) + 3/(x - 5)
5) 4/(x^2 - 4) + 1/(x + 2)
1. **Problem:** Add the following rational expressions:
$$\frac{3}{x+2}+\frac{5}{x-4}$$
2. **Formula to use:** To add rational expressions, first find a common denominator.
Then rewrite each fraction with that denominator before adding the numerators.
The least common denominator is:
$$\left(x+2\right)\left(x-4\right)$$
3. **Rewrite each fraction with the common denominator:**
$$\frac{3}{x+2}=\frac{3\left(x-4\right)}{\left(x+2\right)\left(x-4\right)}$$
$$\frac{5}{x-4}=\frac{5\left(x+2\right)}{\left(x-4\right)\left(x+2\right)}$$
4. **Add the numerators:**
$$\frac{3\left(x-4\right)+5\left(x+2\right)}{\left(x+2\right)\left(x-4\right)}$$
5. **Simplify the numerator:**
$$3\left(x-4\right)+5\left(x+2\right)=3x-12+5x+10$$
$$=8x-2$$
6. **Final answer:**
$$\frac{8x-2}{\left(x+2\right)\left(x-4\right)}$$
You can also factor out a $2$ from the numerator:
$$\frac{2\left(4x-1\right)}{\left(x+2\right)\left(x-4\right)}$$