Question

subtract.
\\(\frac{x - 7}{x + 1} - \frac{5}{x + 6}\\)
simplify your answer as much as possible.

Explanation:

Step1: Find common denominator

The common denominator of $\frac{x-7}{x+1}$ and $\frac{5}{x+6}$ is $(x+1)(x+6)$. Rewrite each fraction with this denominator:
$$\frac{(x-7)(x+6)}{(x+1)(x+6)} - \frac{5(x+1)}{(x+1)(x+6)}$$

Step2: Expand numerators

Expand the products in the numerators:
$$\frac{x^2+6x-7x-42}{(x+1)(x+6)} - \frac{5x+5}{(x+1)(x+6)}$$
Simplify the first numerator:
$$\frac{x^2 - x - 42}{(x+1)(x+6)} - \frac{5x+5}{(x+1)(x+6)}$$

Step3: Combine fractions

Subtract the numerators over the common denominator:
$$\frac{(x^2 - x - 42) - (5x + 5)}{(x+1)(x+6)}$$

Step4: Simplify numerator

Distribute the negative sign and combine like terms:
$$\frac{x^2 - x - 42 - 5x - 5}{(x+1)(x+6)} = \frac{x^2 - 6x - 47}{(x+1)(x+6)}$$

Step5: Expand denominator (optional)

Expand the denominator if needed:
$$\frac{x^2 - 6x - 47}{x^2 + 7x + 6}$$

Answer:

$\frac{x^2 - 6x - 47}{(x+1)(x+6)}$ (or $\frac{x^2 - 6x - 47}{x^2 + 7x + 6}$)