Question

Simplify the complex fraction.
StartStartFraction StartFraction 1 Over y minus 2 EndFraction plus StartFraction 2 Over y plus 4 EndFraction OverOver StartFraction 2 Over y plus 4 EndFraction minus StartFraction 1 Over y minus 5 EndFraction EndEndFraction

Answer 1

To simplify the complex fraction

\[
\frac{\frac{1}{y - 2} + \frac{2}{y + 4}}{\frac{2}{y + 4} - \frac{1}{y - 5}},
\]

we start by simplifying the numerator and the denominator separately.

### Step 1: Simplifying the Numerator

The numerator is

\[
\frac{1}{y - 2} + \frac{2}{y + 4}.
\]

To combine these fractions, we need a common denominator, which is \((y - 2)(y + 4)\). Rewriting each fraction:

\[
\frac{1}{y - 2} = \frac{1(y + 4)}{(y - 2)(y + 4)} = \frac{y + 4}{(y - 2)(y + 4)}
\]

\[
\frac{2}{y + 4} = \frac{2(y - 2)}{(y + 4)(y - 2)} = \frac{2y - 4}{(y - 2)(y + 4)}.
\]

Now, adding these together:

\[
\frac{y + 4 + (2y - 4)}{(y - 2)(y + 4)} = \frac{3y}{(y - 2)(y + 4)}.
\]

### Step 2: Simplifying the Denominator

The denominator is

\[
\frac{2}{y + 4} - \frac{1}{y - 5}.
\]

The common denominator for these two fractions is \((y + 4)(y - 5)\). Rewriting each fraction:

\[
\frac{2}{y + 4} = \frac{2(y - 5)}{(y + 4)(y - 5)} = \frac{2y - 10}{(y + 4)(y - 5)},
\]

\[
\frac{1}{y - 5} = \frac{1(y + 4)}{(y - 5)(y + 4)} = \frac{y + 4}{(y - 5)(y + 4)}.
\]

Now, subtracting these:

\[
\frac{(2y - 10) - (y + 4)}{(y + 4)(y - 5)} = \frac{2y - 10 - y - 4}{(y + 4)(y - 5)} = \frac{y - 14}{(y + 4)(y - 5)}.
\]

### Step 3: Putting it All Together

Now we substitute back into our original complex fraction:

\[
\frac{\frac{3y}{(y - 2)(y + 4)}}{\frac{y - 14}{(y + 4)(y - 5)}}.
\]

This simplifies to:

\[
\frac{3y}{(y - 2)(y + 4)} \times \frac{(y + 4)(y - 5)}{y - 14} = \frac{3y(y - 5)}{(y - 2)(y - 14)}.
\]

### Final Result

Thus, the simplified form of the complex fraction is

\[
\frac{3y(y - 5)}{(y - 2)(y - 14)}.
\]