Final answer:
The simplified expression of (1)/(x+1)-(5x-2)/(x^2-5x-6) is -4/(x-6) after factoring the quadratic, finding the least common denominator, and canceling out common factors.
Step-by-step explanation:
To simplify the expression (1)/(x+1)-(5x-2)/(x^2-5x-6), we first need to factorize the quadratic expression in the denominator of the second term. The quadratic expression x^2 - 5x - 6 can be factored into (x-6)(x+1). By doing this, we identify the least common denominator (LCD) of the given expression as (x-6)(x+1).
Next, we convert each term to have the LCD as its denominator:
- The first term is already over (x+1), so we just need to multiply its numerator and denominator by (x-6).
- The second term is already over (x-6)(x+1), so it remains unchanged.
Once both fractions have the common denominator, we can combine them:
(1(x-6))/(x-6)(x+1) - (5x-2)/(x-6)(x+1)
Simplifying this, we get:
(x-6-5x+2)/(x^2-5x-6)
And further simplification leads to:
(-4x-4)/(x^2-5x-6) or (-4)(x+1)/(x-6)(x+1)
We can now cancel out the common factors to get the final simplified expression:
-4/(x-6)