Mr. Knotts's Subtraction: Spot The Error!

$\frac{x}{x^2-1}-\frac{1}{x-1}$

Step 1: $\frac{x}{(x+1)(x-1)}-\frac{1}{x-1}$

Step 2: $\frac{x}{(x+1)(x-1)}-\frac{1(x+1)}{(x+1)(x-1)}$

Let's break down Mr. Knotts's work step by step to see if we can spot any errors in his calculations. We're essentially trying to simplify the expression and identify if each step he took was mathematically sound. Spotting errors in algebraic manipulations is a fundamental skill in mathematics, and this problem gives us a great opportunity to practice that. So, let's get started!

Analyzing Mr. Knotts's Steps

Step 1: Factoring the Denominator

In Step 1, Mr. Knotts transforms the first term of the expression. He starts with $\frac{x}{x^2-1}$ and rewrites the denominator $x^2 - 1$ as $(x+1)(x-1)$. Guys, this is a classic difference of squares factorization, and it's absolutely correct! The difference of squares formula states that $a^2 - b^2 = (a+b)(a-b)$. In this case, $a = x$ and $b = 1$, so $x^2 - 1^2$ indeed factors to $(x+1)(x-1)$. This step is a standard algebraic technique used to simplify rational expressions, and it allows us to find a common denominator more easily. So far, so good for Mr. Knotts! Factoring the denominator is a crucial move because it sets the stage for combining the two fractions. By expressing the denominator in its factored form, we can clearly see the common factors that will help us find the least common denominator (LCD).

Therefore, $\frac{x}{x^2-1}$ correctly becomes $\frac{x}{(x+1)(x-1)}$. No issues here!

Step 2: Obtaining a Common Denominator

Step 2 is where things get interesting. Mr. Knotts aims to subtract the second term, $\frac{1}{x-1}$, from the first term. To do this, he needs a common denominator. The first term already has the denominator $(x+1)(x-1)$, so he wants to make the second term have the same denominator. To achieve this, he multiplies both the numerator and the denominator of the second term by $(x+1)$. This gives us $\frac{1(x+1)}{(x+1)(x-1)}$, which is equivalent to $\frac{x+1}{(x+1)(x-1)}$. This is a valid algebraic manipulation because multiplying the numerator and denominator of a fraction by the same non-zero expression doesn't change the value of the fraction. It's like multiplying by 1, just in a disguised form. By doing this, Mr. Knotts successfully creates a common denominator, allowing him to combine the two fractions into a single expression.

So, $\frac{1}{x-1}$ becomes $\frac{1(x+1)}{(x+1)(x-1)}$, which simplifies to $\frac{x+1}{(x+1)(x-1)}$. This step is also correct!

Continuing the Calculation (Completing Mr. Knotts's Work)

Now that we've verified that Steps 1 and 2 are correct, let's continue the calculation to see what the final simplified expression should be. This will give us a clear benchmark against which to evaluate Mr. Knotts's complete work, assuming he continued beyond these two steps.

After Step 2, the expression looks like this:

$\frac{x}{(x+1)(x-1)} - \frac{x+1}{(x+1)(x-1)}$

Since we now have a common denominator, we can combine the numerators:

$\frac{x - (x+1)}{(x+1)(x-1)}$

Distribute the negative sign in the numerator:

$\frac{x - x - 1}{(x+1)(x-1)}$

Simplify the numerator:

$\frac{-1}{(x+1)(x-1)}$

Finally, we can rewrite the denominator in its original form:

$\frac{-1}{x^2-1}$

So, the simplified expression is $\frac{-1}{x^2-1}$. This is the correct answer if Mr. Knotts were to complete the problem correctly.

Possible Statements about Mr. Knotts's Work

Based on our analysis, here are some possible statements that could be true about Mr. Knotts's work, along with justification:

1. Mr. Knotts correctly factored the denominator in Step 1. As we discussed, $x^2 - 1$ indeed factors to $(x+1)(x-1)$ using the difference of squares formula. This is a fundamental algebraic identity, and Mr. Knotts applied it correctly.

2. Mr. Knotts correctly obtained a common denominator in Step 2. By multiplying the numerator and denominator of the second term by $(x+1)$, he successfully created a common denominator of $(x+1)(x-1)$, which is essential for subtracting the fractions.

3. If Mr. Knotts stopped at Step 2, his work is incomplete. While Steps 1 and 2 are individually correct, the problem requires finding the difference of the two expressions. This means he needs to combine the fractions and simplify the result, which he hasn't done yet.

4. If Mr. Knotts continued and made an error after Step 2, then his final answer would be incorrect. Common errors in this type of problem include mistakes in distributing the negative sign when combining the numerators or errors in simplifying the resulting expression.

5. If Mr. Knotts continued and correctly simplified after Step 2, then his final answer would be $\frac{-1}{x^2-1}$ As shown above, this is the correct answer.

Choosing the Correct Statement

To determine which statement is true about Mr. Knotts's work, we need to know what he did after Step 2. Since the problem doesn't provide that information, we can only say for sure that Steps 1 and 2 are correct. Therefore, any statement claiming he made an error in those steps would be false. Also, the most accurate statement would be that Mr. Knotts has correctly performed the initial steps of finding the difference of the expressions by factoring and finding the common denominator.

In summary, without further information, the most accurate assessment is that Mr. Knotts demonstrated a correct understanding of factoring and finding common denominators, but the problem remains unfinished at Step 2. To fully evaluate his work, we'd need to see the subsequent steps he took to simplify the expression and arrive at a final answer.