Question
PROBLEM-SOLVING 6 Simplify the following algebraic a \( \frac{5}{x+1}+\frac{2}{x+4} \)
Ask by Schultz Joseph. in Australia
Jan 20,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form is \( \frac{7x+22}{x^{2}+5x+4} \).
Solution
Calculate or simplify the expression \( \frac{5}{x+1}+\frac{2}{x+4} \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{5}{x+1}+\frac{2}{x+4}\)
- step1: Reduce fractions to a common denominator:
\(\frac{5\left(x+4\right)}{\left(x+1\right)\left(x+4\right)}+\frac{2\left(x+1\right)}{\left(x+4\right)\left(x+1\right)}\)
- step2: Rewrite the expression:
\(\frac{5\left(x+4\right)}{\left(x+1\right)\left(x+4\right)}+\frac{2\left(x+1\right)}{\left(x+1\right)\left(x+4\right)}\)
- step3: Transform the expression:
\(\frac{5\left(x+4\right)+2\left(x+1\right)}{\left(x+1\right)\left(x+4\right)}\)
- step4: Multiply the terms:
\(\frac{5x+20+2\left(x+1\right)}{\left(x+1\right)\left(x+4\right)}\)
- step5: Multiply the terms:
\(\frac{5x+20+2x+2}{\left(x+1\right)\left(x+4\right)}\)
- step6: Add the terms:
\(\frac{7x+22}{\left(x+1\right)\left(x+4\right)}\)
- step7: Multiply the terms:
\(\frac{7x+22}{x^{2}+5x+4}\)
The simplified form of the expression \( \frac{5}{x+1}+\frac{2}{x+4} \) is \( \frac{7x+22}{x^{2}+5x+4} \).
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Simplify this solution Bonus Knowledge
To simplify \( \frac{5}{x+1}+\frac{2}{x+4} \), start by finding a common denominator, which is \((x+1)(x+4)\). Rewrite each fraction: \[ \frac{5(x+4)}{(x+1)(x+4)} + \frac{2(x+1)}{(x+1)(x+4)}. \] This gives: \[ \frac{5x + 20 + 2x + 2}{(x+1)(x+4)} = \frac{7x + 22}{(x+1)(x+4)}. \] So, the simplified form is: \[ \frac{7x + 22}{(x+1)(x+4)}. \] Always check if further simplification is possible, but in this case, 7x + 22 and the denominator do not share any common factors.
