Describe the error made in subtracting the two rational expressions shown: \( \frac{1}{x-2}-\frac{1}{x+1} \) \( =\frac{x+1}{(x-2)(x+1)}-\frac{x-2}{(x-2)(x+1)} \) \( =\frac{-1}{(x-2)(x+1)} \)

Let's analyze the subtraction of the two rational expressions step by step to identify the error made. 1. **Original Expression**: \[ \frac{1}{x-2} - \frac{1}{x+1} \] 2. **Finding a Common Denominator**: The common denominator for the two fractions is \((x-2)(x+1)\). 3. **Rewriting Each Fraction**: - The first fraction becomes: \[ \frac{1}{x-2} = \frac{1 \cdot (x+1)}{(x-2)(x+1)} = \frac{x+1}{(x-2)(x+1)} \] - The second fraction becomes: \[ \frac{1}{x+1} = \frac{1 \cdot (x-2)}{(x+1)(x-2)} = \frac{x-2}{(x-2)(x+1)} \] 4. **Subtracting the Two Fractions**: Now we can subtract the two fractions: \[ \frac{x+1}{(x-2)(x+1)} - \frac{x-2}{(x-2)(x+1)} = \frac{(x+1) - (x-2)}{(x-2)(x+1)} \] 5. **Simplifying the Numerator**: The numerator simplifies as follows: \[ (x+1) - (x-2) = x + 1 - x + 2 = 3 \] Therefore, the expression becomes: \[ \frac{3}{(x-2)(x+1)} \] 6. **Identifying the Error**: The error in the original calculation occurs in the step where the numerator was incorrectly simplified. The original claim was: \[ (x+1) - (x-2) = -1 \] This is incorrect; the correct simplification is \(3\), not \(-1\). Thus, the correct result of the subtraction should be: \[ \frac{3}{(x-2)(x+1)} \] In summary, the error was in the simplification of the numerator during the subtraction of the two fractions.