Question
Write the following as a single rational expression \( \frac{1}{x-1}-\frac{1}{x} \)
Ask by Hill Warren. in the United States
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( \frac{1}{x^{2}-x} \).
Solution
Calculate or simplify the expression \( \frac{1}{x-1}-\frac{1}{x} \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{1}{x-1}-\frac{1}{x}\)
- step1: Reduce fractions to a common denominator:
\(\frac{x}{\left(x-1\right)x}-\frac{x-1}{x\left(x-1\right)}\)
- step2: Rewrite the expression:
\(\frac{x}{\left(x-1\right)x}-\frac{x-1}{\left(x-1\right)x}\)
- step3: Transform the expression:
\(\frac{x-\left(x-1\right)}{\left(x-1\right)x}\)
- step4: Subtract the terms:
\(\frac{1}{\left(x-1\right)x}\)
- step5: Multiply the terms:
\(\frac{1}{x^{2}-x}\)
The given expression \( \frac{1}{x-1}-\frac{1}{x} \) can be simplified to \( \frac{1}{x^{2}-x} \).
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Simplify this solution Bonus Knowledge
To combine the two fractions into a single rational expression, we need a common denominator. The denominators here are \( x-1 \) and \( x \). The least common denominator (LCD) is \( x(x-1) \). Now, we can rewrite each fraction: \[ \frac{1}{x-1} = \frac{x}{x(x-1)} \] \[ \frac{1}{x} = \frac{x-1}{x(x-1)} \] Now, we can subtract the two fractions: \[ \frac{x}{x(x-1)} - \frac{x-1}{x(x-1)} = \frac{x - (x-1)}{x(x-1)} = \frac{x - x + 1}{x(x-1)} = \frac{1}{x(x-1)} \] Thus, the single rational expression is: \[ \frac{1}{x(x-1)} \]
