Problem 54 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x^{2}-2 x-24}-\frac{x}{x^{2}-7 x+6}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{-5x}{(x-6)(x+4)(x-1)}\).

Step 1: Factorization

Factor the denominators. The first denominator \(x^{2}-2 x-24\) can be factored as \((x-6)(x+4)\). The second denominator \(x^{2}-7 x+6\) can be factored as \((x-6)(x-1)\).

Step 2: Find common denominator

The common denominator of two fractions is the least common multiple of their denominators. Since the denominators here share the factor \((x-6)\) but have the distinct factors \((x+4)\) and \((x-1)\), the least common multiple is \((x-6)(x+4)(x-1)\).

Step 3: Rewrite fractions with common denominator

Rewrite \(\frac{x}{(x-6)(x+4)}\) and \(\frac{x}{(x-6)(x-1)}\) with \((x-6)(x+4)(x-1)\) as the common denominator. This gives \(\frac{x(x-1)}{(x-6)(x+4)(x-1)}\) and \(\frac{x(x+4)}{(x-6)(x+4)(x-1)}\).

Step 4: Combine the fractions

That will be \(\frac{x(x-1)}{(x-6)(x+4)(x-1)} - \frac{x(x+4)}{(x-6)(x+4)(x-1)}\). Now, since the denominators are the same, we can combine the numerators to simplify this to \(\frac{x(x-1)-x(x+4)}{(x-6)(x+4)(x-1)}\).

Step 5: Simplify the result

Expand the numerators to yield \(\frac{x^{2}-x-x^{2}-4x}{(x-6)(x+4)(x-1)}\), which simplifies to \(\frac{-5x}{(x-6)(x+4)(x-1)}\).