To solve the equation [tex]\(\frac{3}{x+11} - \frac{5}{x-5} = 1\)[/tex], we can follow these steps:
1. Find a common denominator: The common denominator for the fractions [tex]\(\frac{3}{x+11}\)[/tex] and [tex]\(\frac{5}{x-5}\)[/tex] is [tex]\((x+11)(x-5)\)[/tex].
2. Rewrite each term with the common denominator:
[tex]\[
\frac{3}{x+11} = \frac{3(x-5)}{(x+11)(x-5)}
\][/tex]
[tex]\[
\frac{5}{x-5} = \frac{5(x+11)}{(x-5)(x+11)}
\][/tex]
3. Set up the equation with the common denominator:
[tex]\[
\frac{3(x-5) - 5(x+11)}{(x+11)(x-5)} = 1
\][/tex]
4. Combine the numerators and set the equation equal to 1:
[tex]\[
\frac{3(x-5) - 5(x+11)}{(x+11)(x-5)} = 1
\][/tex]
5. Eliminate the denominator by multiplying both sides of the equation by [tex]\((x+11)(x-5)\)[/tex]:
[tex]\[
3(x-5) - 5(x+11) = (x+11)(x-5)
\][/tex]
6. Simplify the left side of the equation:
[tex]\[
3x - 15 - 5x - 55 = (x+11)(x-5)
\][/tex]
Combining like terms:
[tex]\[
-2x - 70 = (x+11)(x-5)
\][/tex]
7. Expand the right side of the equation:
[tex]\[
-2x - 70 = x^2 - 5x + 11x - 55
\][/tex]
Simplifying:
[tex]\[
-2x - 70 = x^2 + 6x - 55
\][/tex]
8. Set up the quadratic equation by moving all terms to one side:
[tex]\[
x^2 + 6x - 55 + 2x + 70 = 0
\][/tex]
Simplifying:
[tex]\[
x^2 + 8x + 15 = 0
\][/tex]
9. Solve the quadratic equation [tex]\(x^2 + 8x + 15 = 0\)[/tex] by factoring:
[tex]\[
(x + 3)(x + 5) = 0
\][/tex]
10. Find the roots of the factored equation:
[tex]\[
x + 3 = 0 \quad \text{or} \quad x + 5 = 0
\][/tex]
Solving these:
[tex]\[
x = -3 \quad \text{or} \quad x = -5
\][/tex]
So, the solutions to the equation [tex]\(\frac{3}{x+11} - \frac{5}{x-5} = 1\)[/tex] are:
[tex]\[ \boxed{-5, -3} \][/tex]
