Question

find the sum:
\frac{6}{x - 4} + \frac{5}{x}
options:
\frac{11}{2x - 4}
\frac{11x - 24}{x^2 - 4x}
\frac{11x - 20}{x^2 - 4x}

find the difference:
\frac{2}{x + 10} - \frac{3}{x + 4}
options:
\frac{-x + 38}{(x + 10)(x + 4)}
\frac{-x - 22}{(x + 10)(x + 4)}
-\frac{1}{6}

Explanation:

First Problem: Find the sum $\frac{6}{x-4}+\frac{5}{x}$

Step1: Find common denominator

The common denominator is $x(x-4)=x^2-4x$.

Step2: Rewrite fractions with LCD

$\frac{6x}{x(x-4)} + \frac{5(x-4)}{x(x-4)}$

Step3: Expand numerator terms

$\frac{6x + 5x - 20}{x^2-4x}$

Step4: Combine like terms

$\frac{11x - 20}{x^2-4x}$

Step1: Find common denominator

The common denominator is $(x+10)(x+4)$.

Step2: Rewrite fractions with LCD

$\frac{2(x+4)}{(x+10)(x+4)} - \frac{3(x+10)}{(x+10)(x+4)}$

Step3: Expand numerator terms

$\frac{2x + 8 - 3x - 30}{(x+10)(x+4)}$

Step4: Combine like terms

$\frac{-x - 22}{(x+10)(x+4)}$

Answer:

$\frac{11x-20}{x^2-4x}$

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Second Problem: Find the difference $\frac{2}{x+10}-\frac{3}{x+4}$