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Chapter 3: Problem 82

What is a rational inequality?

Short Answer

Expert verified
A rational inequality is an inequality that contains a rational expression. Rational expressions are fractions where the numerator and/or the denominator are polynomials. Solving this type of inequality requires finding the critical numbers and testing intervals. Our real solutions are intervals that satisfy the inequality.

Step by step solution

01

Define Rational Inequality

A rational inequality is an inequality that contains a rational expression, a fraction in which the numerator and/or the denominator are polynomials.
02

Point Out Essential Characteristics

These inequalities deal with fractions where the numerator and/or the denominator are polynomials. Like simple fractions, infix rules apply: we cannot divide by zero, which makes understanding undefined values essential.
03

Explain Solving Method

To solve a rational inequality: 1. Simplify the rational expression if it's not already in reduced form. 2. Find the critical numbers by setting the numerator equal to zero, and the denominator as well. (Regarding the denominator, we are looking for values that make the denominator zero as we cannot divide by zero.) 3. Identify the intervals formed by these critical numbers, and test numbers within these intervals in the original inequality.
04

Apply the Concepts

As an example, let's say we have the inequality \(\frac{x-1}{x+2} \geq 0\). The first step would be to find the critical numbers, which are \(x = 1\) and \(x = -2\). We pick a test number from each interval \(-\infty, -2\), \(-2, 1\), and \(1, \infty\) and substitute these numbers back into our inequality. From this, we can find the solution to our inequality.

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Most popular questions from this chapter

Find the slant asymptote of the graph of each rational function and \(\mathbf{b}\). Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$ f(x)=\frac{x^{2}+4}{x} $$

The equations in Exercises \(72-75\) have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$2 x^{3}-15 x^{2}+22 x+15=0 ;[-1,6,1] \text { by }[-50,50,10]$$

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$ 3 x^{2}+16 x<-5 $$

During the 1980 s, the controversial economist Arthur Laffer promoted the idea that tax increases lead to a reduction in government revenue. Called supply- side economics, the theory uses functions such as $$f(x)=\frac{80 x-8000}{x-110}, 30 \leq x \leq 100$$ This function models the government tax revenue, \(f(x),\) in tens of billions of dollars, in terms of the tax rate, \(x\). The graph of the function is shown. It illustrates tax revenue decreasing quite dramatically as the tax rate increases At a tax rate of (gasp) \(100 \%\), the government takes all our money and no one has an incentive to work. With no income earned, zero dollars in tax revenue is generated. CAN'T COPY THE GRAPH a. Find and interpret \(f(40)\). Identify the solution as a point on the graph of the function. b. Rewrite the function by using long division to perform $$(80 x-8000) \div(x-110)$$ Then use this new form of the function to find \(f(40) .\) Do you obtain the same answer as you did in part (a)? c. Is \(f\) a polynomial function? Explain your answer.

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$ f(x)=\frac{x}{x+4} $$
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