Question

what statement is true about the students work?
students work
\\(\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)}\\)
the students work is incorrect. when rewriting the terms with common denominators, the numerators are incorrect.
the students work is incorrect. the common denominator is not \\((x - 2)(x + 1)\\).
the students work is incorrect. the simplified numerator of the difference should be 3, not \\(-1\\).
the students work is correct.

Explanation:

Step1: Analyze the student's work

The student is subtracting two rational expressions: $\frac{1}{x - 2}-\frac{1}{x + 1}$. To subtract them, we need a common denominator, which is $(x - 2)(x + 1)$. Then we rewrite each fraction with this common denominator: $\frac{1\times(x + 1)}{(x - 2)(x + 1)}-\frac{1\times(x - 2)}{(x - 2)(x + 1)}$.

Step2: Simplify the numerators

Now, simplify the numerators: $(x + 1)-(x - 2)=x + 1 - x + 2 = 3$. But the student got $(x + 1)-(x - 2)=x + 1 - x - 2=-1$, which is incorrect. The simplified numerator of the difference should be 3, not -1.

Step3: Evaluate the options

• Option 1: The numerators after getting common denominators are $(x + 1)$ and $(x - 2)$, so this statement is wrong.
• Option 2: The common denominator is correct as $(x - 2)(x + 1)$, so this statement is wrong.
• Option 3: This matches our analysis that the numerator should be 3, not -1.
• Option 4: Since we found a mistake, the student's work is not correct, so this statement is wrong.

Answer:

The student's work is incorrect. The simplified numerator of the difference should be 3, not -1. (The third option among the given statements)