In order to compare Emma, Steve, Maria, and George's solutions and select the student who correctly subtracted the rational expressions, let's work through the problem step-by-step.
The problem requires us to subtract the following rational expressions:
[tex]$
\frac{1}{x-1}-\frac{3}{x^2+2x-3}
$[/tex]
First, we should factor the denominator of the second fraction:
The quadratic expression [tex]\( x^2 + 2x - 3 \)[/tex] can be factored into:
[tex]$
x^2 + 2x - 3 = (x + 3)(x - 1)
$[/tex]
Now, the problem turns into:
[tex]$
\frac{1}{x-1}-\frac{3}{(x+3)(x-1)}
$[/tex]
Next, we need a common denominator for the subtraction. The common denominator for our fractions is [tex]\( (x - 1)(x + 3) \)[/tex].
Rewriting both fractions with this common denominator, we have:
[tex]$
\frac{1}{x-1} = \frac{1 \cdot (x + 3)}{(x - 1)(x + 3)} = \frac{x + 3}{(x - 1)(x + 3)}
$[/tex]
The second fraction is already in a form with the common denominator:
[tex]$
\frac{3}{(x + 3)(x - 1)}
$[/tex]
Now we subtract these fractions:
[tex]$
\frac{x + 3}{(x - 1)(x + 3)} - \frac{3}{(x - 1)(x + 3)}
$[/tex]
Combining the fractions over the common denominator:
[tex]$
\frac{(x + 3) - 3}{(x - 1)(x + 3)}
$[/tex]
Simplify the numerator:
[tex]$
(x + 3) - 3 = x
$[/tex]
So, the expression simplifies to:
[tex]$
\frac{x}{(x - 1)(x + 3)}
$[/tex]
Therefore, the correct simplified form of the expression is:
[tex]$
\frac{x}{(x - 1)(x + 3)}
$[/tex]
Looking at the student choices:
- A. Steve: [tex]\(0\)[/tex]
- B. Emma: [tex]\(\frac{x}{(x-1)(x+3)}\)[/tex]
- C. Maria: [tex]\(\frac{x+3}{(x-1)(x+3)}\)[/tex]
- D. George: [tex]\(\frac{-2}{(x-1)(x+3)}\)[/tex]
The correct solution corresponds to student:
B. Emma
Thus, Emma is the student who correctly subtracted the rational expressions.
