Simplify (2x+1)/(x+4)+(1-x)/(x-3)

Problem

(2*x+1)/(x+4)+(1−x)/(x−3)

Solution

1. Find a common denominator by multiplying the two distinct linear factors in the denominators.

(x+4)*(x−3)

2. Rewrite each fraction with the common denominator by multiplying the numerator and denominator of each term by the missing factor.

((2*x+1)*(x−3))/((x+4)*(x−3))+((1−x)*(x+4))/((x+4)*(x−3))

3. Expand the numerators using the FOIL method (First, Outer, Inner, Last).

(2*x+1)*(x−3)=2*x2−6*x+x−3=2*x2−5*x−3

(1−x)*(x+4)=x+4−x2−4*x=−x2−3*x+4

4. Combine the numerators over the common denominator and group like terms.

((2*x2−5*x−3)+(−x2−3*x+4))/((x+4)*(x−3))

5. Simplify the numerator by adding the coefficients of the x2 x and constant terms.

2*x2−x2=x2

−5*x−3*x=−8*x

−3+4=1

Final Answer

(2*x+1)/(x+4)+(1−x)/(x−3)=(x2−8*x+1)/((x+4)*(x−3))

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