Question
Simplify the expression \( \frac{1}{x+4}-\frac{1}{x+7} \) and give your answer in the form of \( \frac{f(x)}{g(x)} \). Your answer for the function \( f(x) \) is : Your answer for the function \( g(x) \) is :
Ask by Chen Griffiths. in the United States
Mar 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x) = 3 \) and \( g(x) = x^2 + 11x + 28 \).
Solution
Calculate or simplify the expression \( \frac{1}{x+4}-\frac{1}{x+7} \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{1}{x+4}-\frac{1}{x+7}\)
- step1: Reduce fractions to a common denominator:
\(\frac{x+7}{\left(x+4\right)\left(x+7\right)}-\frac{x+4}{\left(x+7\right)\left(x+4\right)}\)
- step2: Rewrite the expression:
\(\frac{x+7}{\left(x+4\right)\left(x+7\right)}-\frac{x+4}{\left(x+4\right)\left(x+7\right)}\)
- step3: Transform the expression:
\(\frac{x+7-\left(x+4\right)}{\left(x+4\right)\left(x+7\right)}\)
- step4: Calculate:
\(\frac{3}{\left(x+4\right)\left(x+7\right)}\)
- step5: Multiply the terms:
\(\frac{3}{x^{2}+11x+28}\)
The simplified expression is \( \frac{3}{x^2+11x+28} \).
Therefore, the function \( f(x) \) is 3 and the function \( g(x) \) is \( x^2+11x+28 \).
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Simplify this solution Beyond the Answer
To simplify the expression \( \frac{1}{x+4}-\frac{1}{x+7} \), we need a common denominator. The common denominator is \( (x+4)(x+7) \). Now, rewrite each fraction: \[ \frac{1}{x+4} = \frac{x+7}{(x+4)(x+7)} \] \[ \frac{1}{x+7} = \frac{x+4}{(x+4)(x+7)} \] Now substitute these into the expression: \[ \frac{x+7}{(x+4)(x+7)} - \frac{x+4}{(x+4)(x+7)} = \frac{(x+7) - (x+4)}{(x+4)(x+7)} \] Simplifying the numerator: \[ (x+7) - (x+4) = 7 - 4 = 3 \] So, the expression becomes: \[ \frac{3}{(x+4)(x+7)} \] Thus, in the form of \( \frac{f(x)}{g(x)} \): Your answer for the function \( f(x) \) is: \( 3 \) Your answer for the function \( g(x) \) is: \( (x+4)(x+7) \)
