To solve the equation [tex]\( y + \frac{y^2-5}{y^2-1} = \frac{y^2+y+2}{y+1} \)[/tex], Janet needs to eliminate the denominators on both sides. Here's a step-by-step explanation of how to determine what to multiply both sides by:
1. Identify the denominators: The denominators in the equation are [tex]\( y^2-1 \)[/tex] and [tex]\( y+1 \)[/tex].
2. Factorize the denominators:
- [tex]\( y^2-1 \)[/tex] can be written as [tex]\( (y-1)(y+1) \)[/tex].
3. Determine the Least Common Multiple (LCM):
- For the denominators [tex]\( y^2-1 = (y-1)(y+1) \)[/tex] and [tex]\( y+1 \)[/tex], the LCM is the expression that contains all the unique factors at their highest power.
- The expression [tex]\( y^2-1 \)[/tex] already includes the factor [tex]\( y+1 \)[/tex], so the LCM of [tex]\( y^2-1 \)[/tex] and [tex]\( y+1 \)[/tex] is [tex]\( y^2-1 \)[/tex].
4. Multiply both sides of the equation by the LCM:
- We need to multiply both sides of the equation by [tex]\( y^2-1 \)[/tex] to clear the denominators and simplify the equation.
Therefore, Janet should multiply both sides of the equation by [tex]\( y^2-1 \)[/tex].
So the correct answer is [tex]\( y^2-1 \)[/tex].
