Question

Solve for the variable x in the equation shown below.
StartFraction 6 Over x plus 5 EndFraction minus StartFraction 9 Over x minus 5 EndFraction equals StartFraction 6 Over x squared minus 25 EndFraction
Question content area bottom
Part 1
A.
​{negative 27​}
B.
​{27​}
C.StartSet StartRoot 73 EndRoot EndSet
StartSet StartRoot 73 EndRoot EndSet
D.
​{81​}

Answer 1

To solve the equation

\[
\frac{6}{x + 5} - \frac{9}{x - 5} = \frac{6}{x^2 - 25}
\]

we start by noticing that \(x^2 - 25\) can be factored as \((x + 5)(x - 5)\). Thus, we can rewrite the equation as:

\[
\frac{6}{x + 5} - \frac{9}{x - 5} = \frac{6}{(x + 5)(x - 5)}
\]

Now, we can find a common denominator for the left side, which is \((x + 5)(x - 5)\):

\[
\frac{6(x - 5) - 9(x + 5)}{(x + 5)(x - 5)} = \frac{6}{(x + 5)(x - 5)}
\]

This gives us the equation:

\[
\frac{6(x - 5) - 9(x + 5)}{(x + 5)(x - 5)} = \frac{6}{(x + 5)(x - 5)}
\]

Now, we can simplify the left-hand side:

\[
6(x - 5) - 9(x + 5) = 6x - 30 - 9x - 45 = -3x - 75
\]

So we have:

\[
\frac{-3x - 75}{(x + 5)(x - 5)} = \frac{6}{(x + 5)(x - 5)}
\]

Since the denominators are equal, we can set the numerators equal to each other:

\[
-3x - 75 = 6
\]

Now, we can solve for \(x\):

\[
-3x = 6 + 75
\]
\[
-3x = 81
\]
\[
x = -27
\]

Thus, the solution for \(x\) is:

\[
\boxed{-27}
\]

Based on the options provided, the answer is:

A. {negative 27}