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Chapter 0: Problem 56

Add or subtract as indicated. $$\frac{x+5}{x^{2}-4}-\frac{x+1}{x-2}$$

Short Answer

Expert verified
\(\frac{-x^{2}-2x+3}{x^{2}-4}\) or \(\frac{-x^{2}-2x+3}{(x+2)(x-2)}\).

Step by step solution

01

Identify the common denominator

Since the two denominators \(x^{2}-4\) and \(x-2\) are not the same, they need to be simplified into a common form. \(x^{2}-4\) is in fact a difference of squares and can be factored into \((x-2)(x+2)\). Therefore, the common denominator of the two fractions is \((x-2)(x+2)\).
02

Write the expressions with the common denominator

Rewrite the expressions with the common denominator. That gives: \(\frac{x+5}{(x+2)(x-2)}-\frac{(x+1)(x+2)}{(x-2)(x+2)}\).
03

Subtract the numerators

Now that the fractions have the same denominator, subtract the numerators: \(\frac{x+5-(x^{2}+3x+2)}{(x+2)(x-2)}\).
04

Simplify the result

Simplify the numerator by combining like terms: \(\frac{-x^{2}-2x+3}{(x+2)(x-2)}\)

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Most popular questions from this chapter

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned} &2>1\\\ &2(y-x)>1(y-x)\\\ &2 y-2 x>y-x\\\ &\begin{aligned} y-2 x &>-x \\ y &>x \end{aligned} \end{aligned}$$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

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