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Question Add the two rational expressions x+3x+4\frac{x+3}{x+4} and x−4x−3\frac{x-4}{x-3}, and simplify the result to lowest terms.

Studdy Solution

STEP 1

Assumptions
1. We are given two rational expressions: x+3x+4\frac{x+3}{x+4} and x−4x−3\frac{x-4}{x-3}.
2. We are asked to perform addition on these two rational expressions.
3. We need to reduce the final answer to its lowest terms.

STEP 2

To add two rational expressions with different denominators, we need to find a common denominator. The common denominator will be the product of the two distinct denominators.
Common denominator=(x+4)(x−3)Common\, denominator = (x+4)(x-3)

STEP 3

Rewrite each fraction with the common denominator.
x+3x+4=(x+3)(x−3)(x+4)(x−3)\frac{x+3}{x+4} = \frac{(x+3)(x-3)}{(x+4)(x-3)} x−4x−3=(x−4)(x+4)(x−3)(x+4)\frac{x-4}{x-3} = \frac{(x-4)(x+4)}{(x-3)(x+4)}

STEP 4

Now, perform the multiplication in the numerators for both expressions.
(x+3)(x−3)(x+4)(x−3)=x2−3x+3x−9(x+4)(x−3)\frac{(x+3)(x-3)}{(x+4)(x-3)} = \frac{x^2-3x+3x-9}{(x+4)(x-3)} (x−4)(x+4)(x−3)(x+4)=x2+4x−4x−16(x−3)(x+4)\frac{(x-4)(x+4)}{(x-3)(x+4)} = \frac{x^2+4x-4x-16}{(x-3)(x+4)}

STEP 5

Simplify the numerators by combining like terms.
x2−3x+3x−9(x+4)(x−3)=x2−9(x+4)(x−3)\frac{x^2-3x+3x-9}{(x+4)(x-3)} = \frac{x^2-9}{(x+4)(x-3)} x2+4x−4x−16(x−3)(x+4)=x2−16(x−3)(x+4)\frac{x^2+4x-4x-16}{(x-3)(x+4)} = \frac{x^2-16}{(x-3)(x+4)}

STEP 6

Now that both fractions have the same denominator, we can add the numerators directly.
x2−9(x+4)(x−3)+x2−16(x−3)(x+4)=(x2−9)+(x2−16)(x+4)(x−3)\frac{x^2-9}{(x+4)(x-3)} + \frac{x^2-16}{(x-3)(x+4)} = \frac{(x^2-9) + (x^2-16)}{(x+4)(x-3)}

STEP 7

Combine the numerators by adding the like terms.
(x2−9)+(x2−16)(x+4)(x−3)=x2−9+x2−16(x+4)(x−3)\frac{(x^2-9) + (x^2-16)}{(x+4)(x-3)} = \frac{x^2-9 + x^2-16}{(x+4)(x-3)}

STEP 8

Combine the terms in the numerator.
x2−9+x2−16(x+4)(x−3)=2x2−25(x+4)(x−3)\frac{x^2-9 + x^2-16}{(x+4)(x-3)} = \frac{2x^2 - 25}{(x+4)(x-3)}

STEP 9

The resulting expression 2x2−25(x+4)(x−3)\frac{2x^2 - 25}{(x+4)(x-3)} is already in its lowest terms, as the numerator and denominator have no common factors other than 1.
Therefore, the final answer is:
2x2−25(x+4)(x−3)\frac{2x^2 - 25}{(x+4)(x-3)}

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