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Perform the indicated operation and simplify the result. Leave your answer in factored form.

[tex]\[
\frac{6}{x-3} - \frac{3}{x+7}
\][/tex]

[tex]\[
\frac{6}{x-3} - \frac{3}{x+7} = \square
\][/tex]

(Simplify your answer. Type your answer in factored form. Use integers or fractions for any numbers.)



Answer :

GinnyAnswer
To simplify the given expression, we perform the indicated operation and leave the answer in factored form.

Given:
[tex]\[ \frac{6}{x-3} - \frac{3}{x+7} \][/tex]

1. Find a common denominator:
The denominators of the fractions are different, so we'll find a common denominator. The common denominator will be the product of both denominators:
[tex]\[ (x - 3)(x + 7) \][/tex]

2. Rewrite each fraction with the common denominator:

[tex]\[ \frac{6}{x-3} = \frac{6(x+7)}{(x-3)(x+7)} \][/tex]

[tex]\[ \frac{3}{x+7} = \frac{3(x-3)}{(x-3)(x+7)} \][/tex]

3. Combine the fractions:

[tex]\[ \frac{6(x+7)}{(x-3)(x+7)} - \frac{3(x-3)}{(x-3)(x+7)} = \frac{6(x+7) - 3(x-3)}{(x-3)(x+7)} \][/tex]

4. Simplify the numerator:

[tex]\[ 6(x + 7) - 3(x - 3) = 6x + 42 - 3x + 9 = 3x + 51 \][/tex]

So the combined and simplified expression is:

[tex]\[ \frac{3x + 51}{(x-3)(x+7)} \][/tex]

5. Factor the numerator:
Notice that [tex]\(3x + 51\)[/tex] can be factored as [tex]\(3(x + 17)\)[/tex]:

[tex]\[ 3(x + 17) \][/tex]

6. Write the final expression in factored form:

So, the simplified and factored form of the expression is:

[tex]\[ \boxed{\frac{3(x + 17)}{(x - 3)(x + 7)}} \][/tex]

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