To simplify the expression [tex]\(\frac{8}{x+2} - \frac{6}{x+5}\)[/tex], we need to combine the two fractions over a common denominator.
1. Identify the common denominator:
The denominators are [tex]\(x+2\)[/tex] and [tex]\(x+5\)[/tex]. The common denominator will be the product of these two expressions: [tex]\((x+2)(x+5)\)[/tex].
2. Rewrite each fraction with the common denominator:
To rewrite [tex]\(\frac{8}{x+2}\)[/tex] with the common denominator [tex]\((x+2)(x+5)\)[/tex], we multiply the numerator and denominator by [tex]\((x+5)\)[/tex]:
[tex]\[
\frac{8}{x+2} = \frac{8(x+5)}{(x+2)(x+5)}
\][/tex]
Similarly, to rewrite [tex]\(\frac{6}{x+5}\)[/tex] with [tex]\((x+2)(x+5)\)[/tex] as the common denominator, we multiply the numerator and denominator by [tex]\((x+2)\)[/tex]:
[tex]\[
\frac{6}{x+5} = \frac{6(x+2)}{(x+2)(x+5)}
\][/tex]
3. Combine the rewritten fractions:
Now, both fractions have the same denominator, so we can subtract the numerators directly:
[tex]\[
\frac{8(x+5)}{(x+2)(x+5)} - \frac{6(x+2)}{(x+2)(x+5)} = \frac{8(x+5) - 6(x+2)}{(x+2)(x+5)}
\][/tex]
4. Simplify the numerator:
Distribute the constants in the numerator:
[tex]\[
8(x+5) - 6(x+2) = 8x + 40 - 6x - 12
\][/tex]
Combine like terms:
[tex]\[
8x + 40 - 6x - 12 = 2x + 28
\][/tex]
5. Express the final simplified form:
The final expression is:
[tex]\[
\frac{2x + 28}{(x+2)(x+5)} = \frac{2(x + 14)}{(x+2)(x+5)}
\][/tex]
So, the simplified form of the expression [tex]\(\frac{8}{x+2} - \frac{6}{x+5}\)[/tex] is:
[tex]\[
\boxed{\frac{2(x + 14)}{(x + 2)(x + 5)}}
\][/tex]
