Simplify (x+2)/(x-1)-2/(x+6)-14/(x^2+5x-6)

Problem

(x+2)/(x−1)−2/(x+6)−14/(x2+5*x−6)

Solution

1. Factor the denominator of the third term by finding two numbers that multiply to −6 and add to 5

x2+5*x−6=(x+6)*(x−1)

2. Identify the least common denominator (LCD) for all three fractions.

LCD=(x−1)*(x+6)

3. Rewrite each fraction using the LCD by multiplying the numerator and denominator by the missing factors.

((x+2)*(x+6))/((x−1)*(x+6))−(2*(x−1))/((x+6)*(x−1))−14/((x−1)*(x+6))

4. Expand the numerators in the expression.

(x2+8*x+12)/((x−1)*(x+6))−(2*x−2)/((x−1)*(x+6))−14/((x−1)*(x+6))

5. Combine the numerators over the common denominator, being careful to distribute the negative sign to the second term.

(x2+8*x+12−(2*x−2)−14)/((x−1)*(x+6))

6. Simplify the resulting numerator by combining like terms.

(x2+8*x+12−2*x+2−14)/((x−1)*(x+6))

(x2+6*x)/((x−1)*(x+6))

7. Factor the numerator to see if any terms cancel with the denominator.

(x*(x+6))/((x−1)*(x+6))

8. Cancel the common factor (x+6) from the numerator and denominator.

x/(x−1)

Final Answer

(x+2)/(x−1)−2/(x+6)−14/(x2+5*x−6)=x/(x−1)

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