Question

simplify.
\\(\dfrac{\dfrac{7}{x + 4} + \dfrac{2}{x - 5}}{\dfrac{9}{x + 4}}\\)

Explanation:

Step1: Simplify the numerator of the complex fraction

First, we need to add the two fractions in the numerator: \(\frac{7}{x + 4}+\frac{2}{x - 5}\). To add these fractions, we find a common denominator, which is \((x + 4)(x - 5)\).

\[

$$\begin{align*} \frac{7}{x + 4}+\frac{2}{x - 5}&=\frac{7(x - 5)}{(x + 4)(x - 5)}+\frac{2(x + 4)}{(x + 4)(x - 5)}\\ &=\frac{7x-35 + 2x + 8}{(x + 4)(x - 5)}\\ &=\frac{9x-27}{(x + 4)(x - 5)}\\ &=\frac{9(x - 3)}{(x + 4)(x - 5)} \end{align*}$$

\]

Step2: Divide by the denominator of the complex fraction

Now, the complex fraction is \(\frac{\frac{9(x - 3)}{(x + 4)(x - 5)}}{\frac{9}{x + 4}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so we have:

\[

$$\begin{align*} \frac{9(x - 3)}{(x + 4)(x - 5)}\times\frac{x + 4}{9}&=\frac{9(x - 3)(x + 4)}{9(x + 4)(x - 5)} \end{align*}$$

\]

Step3: Cancel out common factors

We can cancel out the common factors of \(9\) and \(x + 4\) (assuming \(x
eq - 4\) and \(x
eq5\) to avoid division by zero):

\[
\frac{9(x - 3)(x + 4)}{9(x + 4)(x - 5)}=\frac{x - 3}{x - 5}
\]

Answer:

\(\frac{x - 3}{x - 5}\)