Chapter 5

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{x^{4}+x^{2}+3}{x^{2}+2}>0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{3 x^{2}+3 x-6}{x^{2}-x-12}$$

Use analytic or graphical methods to solve the inequality. $$\sqrt[3]{x+2} \geq \sqrt[3]{x^{2}}$$

Answer each of the following. Consider the expression \(16^{-34}\) (a) Simplify this expression without using a calculator. Give the answer in both decimal and \(\frac{a}{b}\) form. (b) Write two different radical expressions that are equivalent to it, and use your calculator to evaluate them to show that the result is the same as the decimal form you found in part (a). (c) If your calculator has the capability to convert decimal numbers to fractions, use it to verify your results in part (a).

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{(x-1)^{2}}{x^{2}+4}>0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{4 x^{2}+4 x-24}{x^{2}-3 x-10}$$

Use analytic or graphical methods to solve the inequality. $$\sqrt[3]{4 x^{2}+3}<\sqrt[3]{3-x}$$

CONCEPT CHECK In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$ \frac{1}{x^{2}+1}>0 $$ The numerator of the rational expression, 1, is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nonnegative number, \(x^{2},\) and a positive number, 1. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than \(0,\) and this will always be true, the solution set is \((-\infty, \infty)\) Use similar reasoning to solve each inequality. $$\frac{(x-1)^{2}}{x^{2}+4} \leq 0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{9 x^{2}-1}{x^{2}-4}$$

Use analytic or graphical methods to solve the inequality. $$\sqrt{x-1}>x-1$$

Let \(f(x)=\frac{x^{2}-4}{x^{2}-4} .\) Solve each equation or inequality. (a) \(f(x)=0\) (b) \(f(x)<0\) (c) \(f(x)>0\) (d) \(f(x)=1\)

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{16 x^{2}-9}{x^{2}-9}$$

Use analytic or graphical methods to solve the inequality. $$\sqrt{x+5} \leq \frac{3}{4} x$$

Solve each rational inequality by hand. $$\frac{3-2 x}{1+x}<0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x-3)(x+1)}{(x-1)^{2}}$$

Use analytic or graphical methods to solve the inequality. $$\sqrt{-x}<0$$

Solve each rational inequality by hand. $$\frac{3 x-3}{4-2 x} \geq 0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x(x-2)}{(x+3)^{2}}$$

Use analytic or graphical methods to solve the inequality. $$2+\sqrt{3 x}<1$$

Solve each rational inequality by hand. $$\frac{(x+1)(x-2)}{(x+3)}<0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x}{x^{2}-9}$$

Use analytic or graphical methods to solve the inequality. $$\sqrt{-x}-1<0$$

Solve each problem. Wing Size Suppose that the surface area \(S\) of a bird's wings, in square feet, can be modeled by $$S(w)=1.27 w^{2 / 3}$$ where \(w\) is the weight of the bird in pounds. Estimate the surface area of a bird's wings if the bind weighs 4.0 pounds.

Solve each rational inequality by hand. $$\frac{x(x-3)}{x+2} \geq 0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{-5}{2 x+4}$$

Suppose that the wingspan \(L\) in feet of a bird weighing \(W\) pounds is given by $$L=2.43 W^{0.3326}$$ Estimate the wingspan of a bird that weighs 5.2 pounds.

Use analytic or graphical methods to solve the inequality. $$\sqrt{-x}+3>0$$

Solve each rational inequality by hand. $$\frac{(x+1)^{2}}{x-2} \leq 0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{-4}{3 x+9}$$

Solve each problem. The formula \(f(x)=x^{1.5}\) calculates the number of years it would take for a planet to orbit the sun if its average distance from the sun is \(x\) times farther than that of Earth. If there were a planet located 15 times farther from the sun than Earth, how many years would it take for the planet to orbit the sun?

Solve each rational inequality by hand. $$\frac{(x-2)^{2}}{2 x}>0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(3-x)^{2}}{(1-x)(4+x)}$$

A formula for calculating the distance one can see from an airplane to the horizon on a clear day is given by \(f(x)=1.22 x^{0.5},\) where \(x\) is the altitude of the plane in feet and \(f(x)\) is in miles. If a plane is flying at \(30,000\) feet, how far can the pilot see?

Solve each rational inequality by hand. $$\frac{2 x-5}{x^{2}-1} \geq 0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+4)^{2}}{(x-1)(x+5)}$$

Rainbow trout are sensitive to zinc ions in the water. High concentrations are lethal. The average survival times \(x\) in minutes for trout in various concentrations of zinc ions \(y\) in milligrams per liter (mg/L) are listed in the table. $$\begin{array}{|c|c|c|c|c|} \hline x \text { (in minutes) } & 0.5 & 1 & 2 & 3 \\ \hline y \text { (in mg/L) } & 4500 & 1960 & 850 & 525 \end{array}$$ (a) The data can be modeled by $$ f(x)=a x^{b} $$ where \(a\) and \(b\) are constants. Determine \(a\). (Hint: Let \(f(1)=1960 .)\) (b) Estimate \(b\) (c) Evaluate \(f(4)\) and interpret the result.

Solve each rational inequality by hand. $$\frac{5-x}{x^{2}-x-2}<0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{(x+1)^{2}}{(x+2)(x-3)}$$

Insulation workers who were exposed to asbestos and employed before 1960 experienced an increased likelihood of lung cancer. If a group of insulation workers have a cumulative total of \(100,000\) years of work experience, with their first date of employ. ment \(x\) years ago, then the number of lung cancer cases occurring within the group can be modeled by $$N(x)=0.00437 x^{3.2}$$ (Source: Walker, A... Observation and Inference: An Introduction to the Methods of Epidemiology, Epidemiology Resources, Inc.) (a) Calculate \(N(x)\) when \(x=5,10,\) and \(20 .\) What happens to the likelihood of cancer as \(x\) increases? (b) If \(x\) doubles, does the number of cancer cases also double?

Solve each rational inequality by hand. $$\frac{1}{x-3} \leq \frac{5}{x-3}$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{20+6 x-2 x^{2}}{8+6 x-2 x^{2}}$$

One study of the male fiddler crab showed a connection between the weight of its claws and the animal's total body weight. For a crab weighing over 0.75 gram, the weight of its claws can be estimated by $$f(x)=0.445 x^{1.25}$$ The input \(x\) is the weight of the crab in grams, and the output \(f(x)\) is the weight of the claws in grams. Predict the weight of the claws for a crab that weighs 2 grams. (Source: Huxley, J., Problems of Relative Growth, Methuen and Co.; Brown, D. and P. Rothery, Models in Biology: Mathematics, Staristics, and Computing. John Wiley and Sons.)

Solve each rational inequality by hand. $$\frac{3}{2-x}>\frac{x}{2+x}$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{18+6 x-4 x^{2}}{4+6 x+2 x^{2}}$$

The average weight for a man can sometimes be estimated by $$f(x)=0.117 x^{1.7}$$ where \(x\) represents the man's height in inches and \(f(x)\) is his weight in pounds. What is the average weight of a 68 -inch-tall man?

Solve each rational inequality by hand. $$2-\frac{5}{x}+\frac{2}{x^{2}} \geq 0$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}+1}{x+3}$$

According to one model, the rate at which an animal's heart beats varies with its weight. Smaller animals tend to have faster pulses, whereas larger animals tend to have slower pulses. The table lists average pulse rates in beats per minute (bpm) for animals with various weights in pounds (Ib). Use regression (or some other method) to find values for \(a\) and \(b\) so that \(f(x)=a x^{b}\) models these data. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Weight (in Ib) } & 40 & 150 & 400 & 1000 & 2000 \\ \hline \text { Pulse (in bpm) } & 140 & 72 & 44 & 28 & 20 \end{array}$$ (Image cannot copy)

Solve each rational inequality by hand. $$\frac{1}{x-1}+\frac{1}{x+1}>\frac{3}{4}$$

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{2 x^{2}+3}{x-4}$$