Question

1. $\frac{1}{x - 4}-\frac{1}{x}$
2. $\frac{6}{2x - 8}+\frac{x}{x^{2}-16}$
3. $\frac{x + 5}{x^{2}-4}-\frac{x + 1}{x - 2}$
4. $\frac{-8x - 6}{x^{2}-x - 6}-\frac{x + 3}{3 - x}$

Explanation:

13.

Step1: Find a common denominator

The common denominator of $x - 4$ and $x$ is $x(x - 4)$. Rewrite the fractions: $\frac{1}{x - 4}-\frac{1}{x}=\frac{x}{x(x - 4)}-\frac{x - 4}{x(x - 4)}$.

Step2: Subtract the numerators

$\frac{x-(x - 4)}{x(x - 4)}=\frac{x - x+4}{x(x - 4)}=\frac{4}{x(x - 4)}$.

Step1: Factor the denominators

$2x - 8=2(x - 4)$ and $x^{2}-16=(x + 4)(x - 4)$. The common denominator is $2(x + 4)(x - 4)$. Rewrite the fractions: $\frac{6}{2(x - 4)}+\frac{x}{(x + 4)(x - 4)}=\frac{6(x + 4)}{2(x + 4)(x - 4)}+\frac{2x}{2(x + 4)(x - 4)}$.

Step2: Add the numerators

$\frac{6(x + 4)+2x}{2(x + 4)(x - 4)}=\frac{6x+24 + 2x}{2(x + 4)(x - 4)}=\frac{8x + 24}{2(x + 4)(x - 4)}=\frac{4x+12}{(x + 4)(x - 4)}$.

Step1: Factor the denominator

$x^{2}-4=(x + 2)(x - 2)$. The common denominator is $(x + 2)(x - 2)$. Rewrite the fractions: $\frac{x + 5}{(x + 2)(x - 2)}-\frac{x + 1}{x - 2}=\frac{x + 5}{(x + 2)(x - 2)}-\frac{(x + 1)(x + 2)}{(x + 2)(x - 2)}$.

Step2: Expand and subtract the numerators

$\frac{x + 5-(x^{2}+3x + 2)}{(x + 2)(x - 2)}=\frac{x + 5-x^{2}-3x - 2}{(x + 2)(x - 2)}=\frac{-x^{2}-2x + 3}{(x + 2)(x - 2)}$.

Answer:

$\frac{4}{x(x - 4)}$

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