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Chapter 7: Problem 54

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

Short Answer

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

Step by step solution

01

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 \((x-6)(x-1)\).
02

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 \((x-6)(x+4)(x-1)\).
03

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 \(\frac{x(x+4)}{(x-6)(x+4)(x-1)}\).
04

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 \(\frac{x(x-1)-x(x+4)}{(x-6)(x+4)(x-1)}\).
05

Simplify the result

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

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Most popular questions from this chapter

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{8}-\frac{5}{6}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{y+5}+\frac{3}{4 y}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-1}{6}+\frac{x+2}{3}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y-3)(y+2)}{(y+1)(y-4)}-\frac{(y+2)(y+3)}{(y+1)(4-y)}-\frac{(y+5)(y-1)}{(y+1)(4-y)}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y-4}-\frac{4}{4-y}$$
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