Question

Simplify the expression (x+1)/(x+2) + (x-2)/(x-1)

Answer 1

The expression (x+1)/(x+2) + (x-2)/(x-1) simplifies to ((2x^2 - 5))/((x+2)(x-1)) after finding a common denominator and combining like terms.

Step-by-step explanation:

To simplify the expression (x+1)/(x+2) + (x-2)/(x-1), we need to find a common denominator. The denominators are (x+2) and (x-1), which means the common denominator is their product, (x+2)(x-1). Next, we rewrite each fraction with this common denominator:

• ((x+1)(x-1))/((x+2)(x-1)) + ((x-2)(x+2))/((x+2)(x-1))

Now, expand the numerators:

• ((x^2 - x + x - 1))/((x+2)(x-1)) + ((x^2 + 2x - 2x - 4))/((x+2)(x-1))

Simplify the numerators by combining like terms:

• ((x^2 - 1))/((x+2)(x-1)) + ((x^2 - 4))/((x+2)(x-1))

Add the fractions since they now have a common denominator:

• ((x^2 - 1) + (x^2 - 4))/((x+2)(x-1))

Combine like terms in the numerator:

• ((2x^2 - 5))/((x+2)(x-1))

Thus, the simplified expression is ((2x^2 - 5))/((x+2)(x-1)).