Perform the indicated operations and simplify the result. Leave the answer in factored form.
-$$\frac { 6 - x } { x - 2 } - \frac { 2 x - 1 } { 2 - x }$$
A) $\frac { x + 7 } { x - 2 }$
B) $- \frac { x + 5 } { x - 2 }$
C) $\frac { x + 5 } { x - 2 }$
D) $- \frac { x + 7 } { x - 2 }$

Question

Quizplus · Accepted Answer

To subtract the second fraction from the first, we need to find a common denominator. The common denominator is (x - 2)(2 - x) = - (x - 2)(x - 2) = - (x - 2)^2. So, we have:$$\frac { 6 - x } { x - 2 } - \frac { 2 x - 1 } { 2 - x } = \frac { (6 - x)(- (x - 2)) } { - (x - 2)^2 } - \frac { (2 x - 1)(x - 2) } { - (x - 2)^2 }$$Simplifying the numerator of the first fraction, we get:$$- (6 - x)(- (x - 2)) = - (- 6 + 6 x - x^2 + 2 x) = x^2 - 8 x + 6$$Simplifying the numerator of the second fraction, we get:$$(2 x - 1)(x - 2) = 2 x^2 - 4 x - x + 2 = 2 x^2 - 5 x + 2$$So, we have:$$\frac { 6 - x } { x - 2 } - \frac { 2 x - 1 } { 2 - x } = \frac { x^2 - 8 x + 6 } { - (x - 2)^2 } - \frac { 2 x^2 - 5 x + 2 } { - (x - 2)^2 }$$Combining the fractions, we get:$$\frac { x^2 - 8 x + 6 - 2 x^2 + 5 x - 2 } { - (x - 2)^2 } = \frac { - x^2 - 3 x + 4 } { - (x - 2)^2 } = \frac { x^2 + 3 x - 4 } { (x - 2)^2 }$$Factoring the numerator, we get:$$\frac { x^2 + 3 x - 4 } { (x - 2)^2 } = \frac { (x + 4)(x - 1) } { (x - 2)^2 }$$Therefore, the answer is C) $\frac { x + 5 } { x - 2 }$.

Perform the Indicated Operations and Simplify the Result 6−xx−2−2x−12−x\frac { 6 - x } { x - 2 } - \frac { 2 x - 1 } { 2 - x }x−26−x​−2−x2x−1​

Perform the Indicated Operations and Simplify the Result $\frac { 6 - x } { x - 2 } - \frac { 2 x - 1 } { 2 - x }$