Simplify x/(x^2-7x+6)-x/(x^2-2x-24)

Problem

x/(x2−7*x+6)−x/(x2−2*x−24)

Solution

1. Factor the denominators of both rational expressions.

x2−7*x+6=(x−6)*(x−1)

x2−2*x−24=(x−6)*(x+4)

2. Rewrite the expression using the factored denominators.

x/((x−6)*(x−1))−x/((x−6)*(x+4))

3. Find the least common denominator (LCD) for the two fractions.

LCD=(x−6)*(x−1)*(x+4)

4. Adjust each fraction to have the LCD by multiplying the numerator and denominator by the missing factors.

(x*(x+4))/((x−6)*(x−1)*(x+4))−(x*(x−1))/((x−6)*(x+4)*(x−1))

5. Combine the numerators over the common denominator.

(x*(x+4)−x*(x−1))/((x−6)*(x−1)*(x+4))

6. Distribute the x in the numerator.

(x2+4*x−(x2−x))/((x−6)*(x−1)*(x+4))

7. Simplify the numerator by combining like terms.

(x2+4*x−x2+x)/((x−6)*(x−1)*(x+4))

(5*x)/((x−6)*(x−1)*(x+4))

Final Answer

x/(x2−7*x+6)−x/(x2−2*x−24)=(5*x)/((x−6)*(x−1)*(x+4))

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