Problem 54 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{x}{x^{2}-2 x-24}-\frac{x}{x^{2}-7 x+6}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-5x/(x-6)(x+4)(x-1)\)

Step 1: Factor the Denominators

Split the quadratic expressions of each factor into two binomial expressions: \(x^{2}-2x-24=(x-6)(x+4)\) and \(x^{2}-7x+6=(x-6)(x-1)\). So, we rewrite the expression as \[\frac{x}{(x-6)(x+4)}-\frac{x}{(x-6)(x-1)}\].

Step 2: Find the Least Common Denominator

As \(x-6\) is already a common factor, hence the least common denominator (LCD) for the two fractions will be \((x-6)(x+4)(x-1)\).

Step 3: Rewrite the Fractions with the LCD

We rewrite the fractions using the LCD. The first fraction lacks the \((x-1)\) term, and the second fraction lacks the \((x+4)\) term in the denominator. So \[\frac{x}{(x-6)(x+4)}-\frac{x}{(x-6)(x-1)} = \frac{x(x-1)}{(x-6)(x+4)(x-1)}- \frac{x(x+4)}{(x-6)(x+4)(x-1)}\].

Step 4: Subtract the Fractions

Now subtract the fractions: \[\frac{x(x-1)}{(x-6)(x+4)(x-1)}- \frac{x(x+4)}{(x-6)(x+4)(x-1)} = \frac{x^2 - x - x^2 - 4x}{(x-6)(x+4)(x-1)}\] Simplify to get: \(-5x/(x-6)(x+4)(x-1)\)