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Chapter 7: Problem 12

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{7 x-14}{x^{2}-9 x+20}$$

Short Answer

Expert verified
The rational expression is undefined for \(x=4\) and \(x=5\).

Step by step solution

01

Identify the denominator of the rational expression

The denominator of the given rational expression is \(x^{2}-9 x+20\). With this expression, we can now start figuring out the values of \(x\) that make the rational expression undefined.
02

Set the denominator equal to zero, and solve for \(x\)

The rational expression is undefined where the denominator equals zero. Therefore, to find out when the rational expression is undefined, set the denominator equal to zero: \[x^{2}-9 x+20=0\] This equation is a simple quadratic equation, which can be solved by a number of methods. Factoring is simple in this case: \[(x - 4)(x - 5) = 0\] Therefore, the solutions to this are \(x=4\) and \(x=5\).
03

Identify the undefined points

Thus, the given rational expression is undefined for \(x=4\) and \(x=5\).

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Most popular questions from this chapter

Use a proportion to solve each problem. A person's hair length is proportional to the number of years it has been growing. After 2 years, a person's hair grows 8 inches. The longest moustache on record was grown by Kalyan Sain of India. Sain grew his moustache for 17 years. How long was each side of the moustache?

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{5 x-2}{3 x-4}+\frac{2 x-3}{4-3 x}$$

When two people work together to complete a job, describe one factor that can result in more or less time than the time given by the rational equations we have been using.

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

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x+3}{x^{2}-x-30}+\frac{x-2}{30+x-x^{2}}$$
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