Question

janet wants to solve the equation $y + \frac{y^2 - 5}{y^2 - 1} = \frac{y^2 + y + 2}{y + 1}$. what should she multiply both sides of the equation by?

$\bigcirc\\ y$
$\bigcirc\\ y^2 - 1$
$\bigcirc\\ y + 1$
$\bigcirc\\ y^2 + y + 2$

Explanation:

Step1: Factor denominators

Note that $y^2 - 1 = (y+1)(y-1)$.
The denominators are $1$, $(y+1)(y-1)$, and $y+1$.

Step2: Find least common denominator

The least common denominator (LCD) is the product of the unique factors with highest powers: $(y+1)(y-1) = y^2 - 1$.

Step3: Verify LCD eliminates fractions

Multiplying each term by $y^2 -1$ will cancel all denominators:

• $y \times (y^2 -1) = y(y^2 -1)$
• $\frac{y^2 -5}{y^2 -1} \times (y^2 -1) = y^2 -5$
• $\frac{y^2 + y +2}{y+1} \times (y^2 -1) = (y^2 + y +2)(y-1)$

Answer:

B. $y^2 - 1$