To simplify the expression [tex]\(\frac{5 - \frac{x+2}{x-2}}{x+3}\)[/tex], let's follow the steps methodically:
1. Combine the fractions in the numerator:
[tex]\[
\frac{5 - \frac{x+2}{x-2}}{x+3}
\][/tex]
To combine these, we first need a common denominator for the terms in the numerator.
2. Express [tex]\(5\)[/tex] as a fraction with a common denominator of [tex]\(x-2\)[/tex]:
[tex]\[
5 = \frac{5(x-2)}{x-2}
\][/tex]
Now the expression becomes:
[tex]\[
\frac{\frac{5(x-2) - (x+2)}{x-2}}{x+3}
\][/tex]
3. Simplify the expression inside the numerator:
[tex]\[
5(x-2) - (x+2) = 5x - 10 - x - 2 = 4x - 12
\][/tex]
Therefore,
[tex]\[
\frac{\frac{4x - 12}{x-2}}{x+3}
\][/tex]
4. Simplify the complex fraction by dividing by [tex]\(x+3\)[/tex]:
[tex]\[
\frac{4x-12}{(x-2)(x+3)}
\][/tex]
5. Factor the numerator if possible:
[tex]\[
4x - 12 = 4(x - 3)
\][/tex]
So, the expression now becomes:
[tex]\[
\frac{4(x-3)}{(x-2)(x+3)}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{5 - \frac{x+2}{x-2}}{x+3}\)[/tex] is:
[tex]\[
\frac{4(x-3)}{(x-2)(x+3)}
\][/tex]
