Certainly! Let's perform the indicated operation and simplify the expression \(\frac{6}{x-3} - \frac{2}{x+8}\). Here’s a step-by-step guide to solve and simplify it.
1. Find a common denominator: The denominators in the given expression are \(x-3\) and \(x+8\). The common denominator will be their product: \((x-3)(x+8)\).
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{6}{x-3} = \frac{6(x+8)}{(x-3)(x+8)}
\][/tex]
[tex]\[
\frac{2}{x+8} = \frac{2(x-3)}{(x-3)(x+8)}
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{6(x+8)}{(x-3)(x+8)} - \frac{2(x-3)}{(x-3)(x+8)}
\][/tex]
4. Subtract the numerators:
[tex]\[
\frac{6(x+8) - 2(x-3)}{(x-3)(x+8)}
\][/tex]
5. Distribute and simplify the numerator:
- Distribute 6 in the first term: \(6(x+8) = 6x + 48\).
- Distribute -2 in the second term: \(2(x-3) = 2x - 6\).
Thus, the numerator becomes:
[tex]\[
6x + 48 - 2x + 6
\][/tex]
6. Combine like terms in the numerator:
[tex]\[
(6x - 2x) + (48 + 6) = 4x + 54
\][/tex]
7. Factor out the common factor in the numerator:
[tex]\[
4x + 54 = 2(2x + 27)
\][/tex]
8. Write the simplified fraction:
[tex]\[
\frac{2(2x + 27)}{(x-3)(x+8)}
\][/tex]
So, the simplified form of the given operation is:
[tex]\[
\frac{2(2x + 27)}{(x - 3)(x + 8)}
\][/tex]
This is the expression in its fully factored and simplified form.
