To identify and correct the error made in subtracting the two rational expressions [tex]\(\frac{1}{x-2} - \frac{1}{x+1}\)[/tex], let's go through the correct process step-by-step.
1. Find a common denominator:
The common denominator for the expressions [tex]\(\frac{1}{x-2}\)[/tex] and [tex]\(\frac{1}{x+1}\)[/tex] is [tex]\((x-2)(x+1)\)[/tex].
2. Rewrite each fraction with the common denominator:
[tex]\[\frac{1}{x-2} = \frac{1 \cdot (x+1)}{(x-2)(x+1)} = \frac{x+1}{(x-2)(x+1)}\][/tex]
[tex]\[\frac{1}{x+1} = \frac{1 \cdot (x-2)}{(x-2)(x+1)} = \frac{x-2}{(x-2)(x+1)}\][/tex]
3. Subtract the two fractions:
[tex]\[
\frac{x+1}{(x-2)(x+1)} - \frac{x-2}{(x-2)(x+1)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
= \frac{(x+1) - (x-2)}{(x-2)(x+1)}
\][/tex]
4. Simplify the numerator:
[tex]\[
(x+1) - (x-2) = x + 1 - x + 2 = 1 + 2 = 3
\][/tex]
5. Write the final simplified expression:
[tex]\[
\frac{3}{(x-2)(x+1)}
\][/tex]
Now let's compare this solution to the error described in the original problem statement:
### Original Problem Steps:
1. Find a common denominator:
That step is correct.
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{1}{x-2} = \frac{x+1}{(x-2)(x+1)}, \quad \frac{1}{x+1} = \frac{x-2}{(x-2)(x+1)}
\][/tex]
This step does not reflect rewriting the fractions correctly, but the form is correct.
3. Subtract the fractions:
The critical step is:
[tex]\[
\frac{x+1}{(x-2)(x+1)} - \frac{x-2}{(x-2)(x+1)}
\][/tex]
The error occurs in simplifying the numerator:
[tex]\[
\frac{(x+1) - (x-2)}{(x-2)(x+1)} = \frac{-1}{(x-2)(x+1)}
\][/tex]
Instead of [tex]\((x + 1) - (x - 2)\)[/tex], simplifying it properly gives:
[tex]\[
(x + 1) - (x - 2) = 3
\][/tex]
### Correct Simplification:
The correct numerator obtained is [tex]\(3\)[/tex], not [tex]\(-1\)[/tex].
Thus, the correct final answer should be:
[tex]\[
\frac{3}{(x-2)(x+1)}
\][/tex]
This completes the error identification and the correct working process for subtracting the two rational expressions.
