Chapter 1

Mixture Problem A merchant blends tea that sells for \(\$ 3.00\) a pound with tea that sells for \(\$ 2.75\) a pound to produce 80 lb of a mixture that sells for \(\$ 2.90\) a pound. How many pounds of each type of tea does the merchant use in the blend?

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \frac{2+\sqrt{-8}}{1+\sqrt{-2}} $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ (x-2)^{2}(x-3)(x+1) \leq 0 $$

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ x^{2}-0.011 x-0.064=0 $$

\(5-60\) Find all real solutions of the equation. $$ \frac{1}{x^{3}}+\frac{4}{x^{2}}+\frac{4}{x}=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}-24=0\)

Sharing a Job Candy and Tim share a paper route. It takes Candy 70 \(\mathrm{min}\) to deliver all the papers, and it takes Tim 80 \(\mathrm{min}\) . How long does it take the two when they work together?

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \frac{1-\sqrt{-1}}{1+\sqrt{-1}} $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}\left(x^{2}-1\right) \geq 0 $$

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ x^{2}-2.450 x+1.500=0 $$

\(5-60\) Find all real solutions of the equation. $$ 4 X^{-4}-16 X^{-2}+4=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}-7=0\)

Sharing a Job Stan and Hilda can mow the lawn in 40 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone?

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \frac{\sqrt{-36}}{\sqrt{-2} \sqrt{-9}} $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{3}-4 x>0 $$

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ x^{2}-2.450 x+1.501=0 $$

\(5-60\) Find all real solutions of the equation. $$ \sqrt{\sqrt{x+5}+x}=5 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(8 x^{2}-64=0\)

Sharing a Job Betty and Karen have been hired to paint the houses in a new development. Working together, the women can paint a house in two-thirds the time that it takes Karen working alone. Betty takes 6 h to paint a house alone. How long does it take Karen to paint a house working alone?

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \frac{\sqrt{-7} \sqrt{-49}}{\sqrt{28}} $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ 16 x \leq x^{3} $$

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ x^{2}-1.800 x+0.810=0 $$

\(5-60\) Find all real solutions of the equation. $$ \sqrt[3]{4 x^{2}-4 x}=x $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(5 x^{2}-125=0\)

Sharing a Job Next-door neighbors Bob and Jim use hoses from both houses to fill Bob's swimming pool. They know that it takes 18 h using both hoses. They also know that Bob's hose, used alone, takes 20\(\%\) less time than Jim's hose alone. How much time is required to fill the pool by each hose alone?

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}+49=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x-3}{x+1} \geq 0 $$

Thickness of a Laminate A company manufactures industrial laminates (thin nylon-based sheets) of thickness 0.020 in, with a tolerance of 0.003 in. (a) Find an inequality involving absolute values that describes the range of possible thickness for the laminate. (b) Solve the inequality that you found in part (a).

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ 2.232 x^{2}-4.112 x=6.219 $$

\(5-60\) Find all real solutions of the equation. $$ x^{2} \sqrt{x+3}=(x+3)^{3 / 2} $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}+16=0\)

Distance, Speed, and Time Wendy took a trip from Davenport to Omaha, a distance of 300 \(\mathrm{mi}\) . She traveled part of the way by bus, which arrived at the train station just in time for Wendy to complete her journey by train. The bus averaged \(40 \mathrm{mi} / \mathrm{h},\) and the train averaged 60 \(\mathrm{mi} / \mathrm{h}\) . The entire trip took \(5 \frac{1}{2} \mathrm{h} .\) How long did Wendy spend on the train?

Find all solutions of the equation, and express them in the form \(a+b i\) $$ 9 x^{2}+4=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{2 x+6}{x-2}<0 $$

Range of Height The average height of adult males is \(68.2 \mathrm{in.},\) and 95\(\%\) of adult males have height \(h\) that satisfies the inequality $$ \left|\frac{h-68.2}{2.9}\right| \leq 2 $$ Solve the inequality to find the range of heights.

Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ 12.714 x^{2}+7.103 x=0.987 $$

\(5-60\) Find all real solutions of the equation. $$ \sqrt{11-x^{2}}-\frac{2}{\sqrt{11-x^{2}}}=1 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(6 x^{2}+100=0\)

Distance, Speed, and Time Two cyclists, 90 mi apart, start riding toward each other at the same time. One cycles twice as fast as the other. If they meet 2 h later, at what aver- age speed is each cyclist traveling?

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}-4 x+5=0 $$

Using Distances to Solve Absolute Value Inequalities Recall that \(|a-b|\) is the distance between a and \(b\) on the number line. For any number \(x\) , what do \(|x-1|\) and \(|x-3|\) represent? Use this interpretation to solve the inequality \(|x-1|<|x-3|\) geometrically. In general, if \(a

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{4 x}{2 x+3}>2 $$

Solve the equation for the indicated variable. $$ h=\frac{1}{2} g t^{2}+v_{0} t ; \text { for } t $$

\(5-60\) Find all real solutions of the equation. $$ \sqrt{x+\sqrt{x+2}}=2 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \((x+2)^{2}=4\)

Distance, Speed, and Time A pilot flew a jet from Montreal to Los Angeles, a distance of 2500 \(\mathrm{mi}\) . On the return trip, the average speed was 20\(\%\) faster than the outbound speed. The round-trip took 9 \(\mathrm{h} 10 \mathrm{min}\) . What was the speed from Montreal to Los Angeles?

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}+2 x+2=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ -2<\frac{x+1}{x-3} $$

Solve the equation for the indicated variable. $$ S=\frac{n(n+1)}{2} ; \text { for } n $$

\(5-60\) Find all real solutions of the equation. $$ \sqrt{1+\sqrt{x+\sqrt{2 x+1}}}=\sqrt{5+\sqrt{x}} $$