Question

LaTeX: \frac{9}{x+11}-\frac{6}{x-11}=\frac{6}{x^2-121}

Answer 1

To solve the equation

\[
\frac{9}{x+11} - \frac{6}{x-11} = \frac{6}{x^2 - 121},
\]

we start by recognizing that \( x^2 - 121 \) can be factored:

\[
x^2 - 121 = (x - 11)(x + 11).
\]

This allows us to rewrite the equation as:

\[
\frac{9}{x+11} - \frac{6}{x-11} = \frac{6}{(x-11)(x+11)}.
\]

Next, we find a common denominator for the left-hand side. The common denominator for the fractions is \((x + 11)(x - 11)\). Therefore, we rewrite the left-hand side:

\[
\frac{9(x - 11) - 6(x + 11)}{(x + 11)(x - 11)}.
\]

Now, simplifying the numerator:

\[
9(x - 11) - 6(x + 11) = 9x - 99 - 6x - 66 = 3x - 165.
\]

Thus, we have:

\[
\frac{3x - 165}{(x + 11)(x - 11)} = \frac{6}{(x + 11)(x - 11)}.
\]

Now we equate the numerators:

\[
3x - 165 = 6.
\]

Adding \( 165 \) to both sides gives:

\[
3x = 171.
\]

Dividing by \( 3 \) yields:

\[
x = 57.
\]

Finally, we check for any restrictions. The original denominators must not equal zero:

- \( x + 11 \neq 0 \Rightarrow x \neq -11 \),
- \( x - 11 \neq 0 \Rightarrow x \neq 11 \),
- \( x^2 - 121 \neq 0 \Rightarrow x \neq 11 \text{ and } x \neq -11 \).

Since \( x = 57 \) does not violate any of these restrictions, it is a valid solution.

Thus, the solution is:

\[
\boxed{57}.
\]