Problem 90 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{7 y-2}{y^{2}-y-12}+\frac{2 y}{4-y}+\frac{y+1}{y+3}$$

Step-by-Step Solution

Answer

The simplified result of the operation is \(\frac{9 y - 6}{(y - 4)(y + 3)}\)

Step 1: Factor the Quadratic Polynomial

Factor the quadratic polynomial \(y^{2}-y-12\) in the denominator of the first fraction to obtain \((y-4)(y+3)\).

Step 2: Rewrite the Denominators

Rewrite the denominator of the second fraction to match the standard form \(y-n\). So, \(4-y\) becomes \(-(y-4)\). Rewrite the denominator of the third fraction as \((y+3)\).

Step 3: Find the Common Denominator

Identify the common denominator for the three fractions, which would be \(-(y-4)(y+3)\).

Step 4: Rewrite the Numerators

Rewrite the numerators by multiplying by the necessary factors such that all fractions have the common denominator. For the second fraction, a negative is introduced due to the manipulation in step 2.

Step 5: Perform the Addition

Perform the addition by combining the numerators over the common denominator.

Step 6: Simplify the Result

Simplify the resultant fraction and eliminate any common factor in the numerator and denominator if available.