Question
Express as a single fraction \( \frac{1}{x+1}+\frac{4}{x-2} \)
Ask by Burns Schultz. in South Africa
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( \frac{5x+2}{x^{2}-x-2} \).
Solution
Calculate or simplify the expression \( \frac{1}{x+1}+\frac{4}{x-2} \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{1}{x+1}+\frac{4}{x-2}\)
- step1: Reduce fractions to a common denominator:
\(\frac{x-2}{\left(x+1\right)\left(x-2\right)}+\frac{4\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}\)
- step2: Rewrite the expression:
\(\frac{x-2}{\left(x+1\right)\left(x-2\right)}+\frac{4\left(x+1\right)}{\left(x+1\right)\left(x-2\right)}\)
- step3: Transform the expression:
\(\frac{x-2+4\left(x+1\right)}{\left(x+1\right)\left(x-2\right)}\)
- step4: Multiply the terms:
\(\frac{x-2+4x+4}{\left(x+1\right)\left(x-2\right)}\)
- step5: Calculate:
\(\frac{5x+2}{\left(x+1\right)\left(x-2\right)}\)
- step6: Multiply the terms:
\(\frac{5x+2}{x^{2}-x-2}\)
The expression \( \frac{1}{x+1}+\frac{4}{x-2} \) can be simplified to \( \frac{5x+2}{x^{2}-x-2} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Bonus Knowledge
To combine the fractions \( \frac{1}{x+1}+\frac{4}{x-2} \) into a single fraction, we first need a common denominator. The common denominator is \((x+1)(x-2)\). Now, we can rewrite each fraction with the common denominator: \[ \frac{1}{x+1} = \frac{1(x-2)}{(x+1)(x-2)} = \frac{x-2}{(x+1)(x-2)} \] \[ \frac{4}{x-2} = \frac{4(x+1)}{(x-2)(x+1)} = \frac{4x + 4}{(x+1)(x-2)} \] Now we can add them together: \[ \frac{x-2}{(x+1)(x-2)} + \frac{4x + 4}{(x+1)(x-2)} = \frac{x - 2 + 4x + 4}{(x+1)(x-2)} = \frac{5x + 2}{(x+1)(x-2)} \] So the expression \( \frac{1}{x+1}+\frac{4}{x-2} \) as a single fraction is: \[ \frac{5x + 2}{(x+1)(x-2)} \]
