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Chapter 7: Problem 33

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x+5}{x^{2}-25}$$

Short Answer

Expert verified
The simplified form of the rational expression is \(\frac{1}{x-5}\) for \(x \neq -5\).

Step by step solution

01

Factorize the denominator

To simplify the given rational expression it's necessary to factorize the denominator. The expression \(x^{2}-25\) is a difference of two squares, which can be factored into \((x-5)(x+5)\).
02

Simplify the expression

After the factorization of the denominator, the expression becomes \(\frac{x+5}{(x-5)(x+5)}\). Since \(x+5\) is common in the numerator and the denominator, those can be cancelled out. However, it's necessary to remember that \(x \neq -5\) because that would make the denominator equal to zero, which is undefined in the real number system.
03

Write the simplified expression

After simplification, the resulting expression is \(\frac{1}{x-5}\).

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Most popular questions from this chapter

Graph: \(y=-\frac{2}{3} x+4 .\) (Section 3.4, Example 3)

After adding \(\frac{3 x+1}{4}\) and \(\frac{x+2}{4},\) I simplified the sum by dividing the numerator and the denominator by 4 I use similar procedures to find each of the following sums: $$ \frac{3}{8}+\frac{1}{8} \text { and } \frac{x}{x^{2}-1}+\frac{1}{x^{2}-1} $$

Will help you prepare for the material covered in the next section. a. If \(y=k x\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6}{x-5}+\frac{2}{5-x}$$

perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{2}+\frac{2}{3}$$
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