Question

riley makes a mistake in step 2 while doing her homework. what was her mistake?
\\(\frac{x}{x^2 - 5x + 6} + \frac{3}{x + 3}\\)
step 1: \\(\frac{x}{(x - 2)(x - 3)} + \frac{3}{x + 3}\\)
step 2: \\(\frac{x}{(x - 2)(x + 3)} + \frac{3(x - 2)}{(x - 2)(x + 3)}\\)
step 3: \\(\frac{x + 3x - 6}{(x - 2)(x + 3)}\\)
step 4: \\(\frac{4x - 6}{(x - 2)(x + 3)}\\)

Explanation:

Step1: Identify correct common denominator

The original first fraction has denominator $(x-2)(x-3)$, and the second has $(x+3)$. The correct common denominator is $(x-2)(x-3)(x+3)$.

Step2: Analyze Riley's Step 2 error

Riley incorrectly changed the denominator of the first fraction to $(x-2)(x+3)$ instead of the full common denominator, and only multiplied the second fraction by $(x-2)$ instead of $(x-2)(x-3)$. She altered the first fraction's denominator without adjusting its numerator to match, and used the wrong factor to convert the second fraction to a common denominator.

Answer:

Riley made two related mistakes in Step 2:

1. She incorrectly modified the denominator of the first fraction $\frac{x}{(x-2)(x-3)}$ to $\frac{x}{(x-2)(x+3)}$ without adjusting the numerator to maintain the value of the fraction.
2. She used the wrong common denominator: the correct common denominator for $\frac{x}{(x-2)(x-3)}$ and $\frac{3}{x+3}$ is $(x-2)(x-3)(x+3)$, so the second fraction should be multiplied by $\frac{(x-2)(x-3)}{(x-2)(x-3)}$, not just $\frac{(x-2)}{(x-2)}$.