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Chapter 3: Problem 82

What is a rational inequality?

Short Answer

Expert verified
A rational inequality is an inequality with a rational expression, which is an expression of the form \(f(x) / g(x)\), where \(f(x)\) and \(g(x)\) are polynomials and \(g(x) \neq 0\). When solving rational inequalities, one must factor, identify the zeros of the numerator and denominator, divide the number line into intervals using these zeros, test each interval in the inequality, and express the solution in interval notation, excluding denominator zeros.

Step by step solution

01

Definition of Rational Inequality

A rational inequality is one that contains a rational expression. A rational expression is one that contains a variable in its denominator, and often in its numerator as well. It is of the form \(f(x) / g(x)\), where \(f(x)\) and \(g(x)\) are polynomial functions, and \(g(x) \neq 0\).
02

How It Differs From Ordinary Inequalities

Unlike ordinary inequalities, rational inequalities take into account the domain of the expressions involved. The denominator of a rational expression can never be zero, so when solving rational inequalities, it's crucial to consider values of the variable that make the denominator zero and generally to exclude them from the solution set.
03

Solving Rational Inequalities

To solve a rational inequality, one typically follows these steps: First, factor both the numerator and the denominator of the rational expression. Next, identify values of the variable that make the numerator and the denominator zero. Using these values, divide the number line into intervals. Test each interval in the inequality to determine whether that interval is part of the solution set. Finally, express the solution set in interval notation, excluding any values that make the denominator zero.

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Most popular questions from this chapter

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has a vertical asymptote given by x=3, a horizontal asymptote y=0, y -intercept at -1, and no x -intercept.

In Exercises 104鈥�107, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. There is more than one third-degree polynomial function with the same three \(x\) -intercepts.

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{x-\frac{1}{x}}{x+\frac{1}{x}} $$

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations $$ \text { of } f(x)=\frac{1}{x} \text { to graph } g $$. $$ g(x)=\frac{2 x+7}{x+3} $$
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