Question

$$\frac{5}{x - 5} + \frac{7}{x + 6}$$ select the correct choice below and fill in any answer boxes within your choice. (simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) \\(\bigcirc\\) a. \\(\frac{5}{x - 5} + \frac{7}{x + 6} = \square, x \
eq \square\\) \\(\bigcirc\\) b. \\(\frac{5}{x - 5} + \frac{7}{x + 6} = \square\\), no numbers must be excluded.

Explanation:

Step1: Find a common denominator

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

Step2: Add the fractions

Add the numerators together:
\[

$$\begin{align*} \frac{5(x + 6)+7(x - 5)}{(x - 5)(x + 6)}&=\frac{5x+30 + 7x-35}{(x - 5)(x + 6)}\\ &=\frac{(5x + 7x)+(30 - 35)}{(x - 5)(x + 6)}\\ &=\frac{12x-5}{(x - 5)(x + 6)} \end{align*}$$

\]

Step3: Determine excluded values

The denominators \(x - 5\) and \(x + 6\) cannot be zero. So \(x-5
eq0\) implies \(x
eq5\) and \(x + 6
eq0\) implies \(x
eq - 6\). But in the options, we need to check the form. Option A has a single excluded value box, but actually there are two, but maybe the problem expects us to note that \(x
eq5\) and \(x
eq - 6\), but looking at the option A's format, maybe it's a typo or we proceed with the simplification. The simplified form is \(\frac{12x - 5}{(x - 5)(x + 6)}\) and the excluded values are \(x
eq5\) and \(x
eq - 6\), but since option A has one box for excluded, maybe we consider the main excluded from the denominators. But let's follow the steps.

Answer:

A. \(\frac{5}{x - 5}+\frac{7}{x + 6}=\frac{12x - 5}{(x - 5)(x + 6)}\), \(x
eq5,-6\) (but since the option has one box for \(x
eq\), maybe the problem expects \(x
eq5\) or \(x
eq - 6\), but strictly, the simplified expression is \(\frac{12x - 5}{(x - 5)(x + 6)}\) and excluded values are \(x
eq5\) and \(x
eq - 6\). However, if we follow the option's format, we can write \(\frac{12x - 5}{(x - 5)(x + 6)}\) in the first box and \(5\) (or \(-6\)) in the second, but actually the correct excluded values are \(5\) and \(-6\). But based on the calculation, the simplified expression is \(\frac{12x - 5}{(x - 5)(x + 6)}\) and \(x
eq5\) (and \(x
eq - 6\)).