Question

combine and simplify.
$27 + \frac{11}{x + 4}$
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1. - / 1 points

combine and simplify.
$\frac{x}{x + 3} - \frac{8}{x - 2}$
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Explanation:

First Problem: \( 27 + \frac{11}{x + 4} \)

Step1: Find a common denominator

To combine the terms, we need a common denominator. The first term \( 27 \) can be written as \( \frac{27(x + 4)}{x + 4} \) so that both terms have the denominator \( x + 4 \).
\[
27=\frac{27(x + 4)}{x + 4}
\]

Step2: Add the fractions

Now we add the two fractions:
\[
\frac{27(x + 4)}{x + 4}+\frac{11}{x + 4}=\frac{27(x + 4)+11}{x + 4}
\]

Step3: Expand and simplify the numerator

Expand \( 27(x + 4) \):
\[
27(x + 4)=27x+108
\]
Then add \( 11 \) to it:
\[
27x + 108+11=27x+119
\]
So the simplified form is:
\[
\frac{27x + 119}{x + 4}
\]

Step1: Find a common denominator

The common denominator of \( (x + 3) \) and \( (x - 2) \) is \( (x + 3)(x - 2) \). Rewrite each fraction with this common denominator:
\[
\frac{x}{x + 3}=\frac{x(x - 2)}{(x + 3)(x - 2)}
\]
\[
\frac{8}{x - 2}=\frac{8(x + 3)}{(x + 3)(x - 2)}
\]

Step2: Subtract the fractions

Now subtract the two fractions:
\[
\frac{x(x - 2)}{(x + 3)(x - 2)}-\frac{8(x + 3)}{(x + 3)(x - 2)}=\frac{x(x - 2)-8(x + 3)}{(x + 3)(x - 2)}
\]

Step3: Expand and simplify the numerator

Expand \( x(x - 2) \) and \( 8(x + 3) \):
\[
x(x - 2)=x^{2}-2x
\]
\[
8(x + 3)=8x + 24
\]
Now subtract:
\[
x^{2}-2x-(8x + 24)=x^{2}-2x - 8x-24=x^{2}-10x - 24
\]
So the simplified form is:
\[
\frac{x^{2}-10x - 24}{(x + 3)(x - 2)}
\]
(We can also factor the numerator if possible. Let's check if \( x^{2}-10x - 24 \) can be factored. We need two numbers that multiply to \( - 24 \) and add to \( - 10 \). The numbers are \( -12 \) and \( 2 \). So \( x^{2}-10x - 24=(x - 12)(x + 2) \). So the simplified form can also be written as \( \frac{(x - 12)(x + 2)}{(x + 3)(x - 2)} \))

Answer:

\( \frac{27x + 119}{x + 4} \)

Second Problem: \( \frac{x}{x + 3}-\frac{8}{x - 2} \)