Chapter 1

Track Event Speeds At the 2008 Summer Olympics in Beijing. China, Usain Bolt (Jamaica) set a new Olympic and world record in the 100 -m dash with a time of 9.69 sec. If this pace could be maintained for an entire 26 -mi marathon, what would his time be? How would this time compare to the fastest time for a marathon, which is 2 hr, 3 min, 59 sec, also set in \(2008 ?\) (Hint: 1 m es 3.281 ft.) (Source: World Almanac and Book of Facts.)

The equation \(|7 x+3|=-5 x\) cannot have a positive solution. Why?

A manufacturing firm wants to package its product in a cylindrical container 3 ft high with surface area \(8 \pi \mathrm{ft}^{2} .\) What should the radius of the circular top and bottom of the container be? (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.)

Break-Even Interval Find all intervals where each product will at least break even. The cost to produce \(x\) units of picture frames is \(C=50 x+5000\), while the revenue "is \(R=60 x\).

Solve each equation. $$\frac{2 x}{x-2}=5+\frac{4 x^{2}}{x-2}$$

Solve each equation. $$0.5 x+\frac{4}{3} x=x+10$$

Solve each equation by the square root property. $$x^{2}=16$$

Solve each equation. $$\frac{2}{3} x+0.25 x=x+2$$

Break-Even Interval Find all intervals where each product will at least break even. The cost to produce \(x\) units of baseball caps is \(C=100 x+6000\), while the revenue is \(R=500 x\)?

Solve each equation. $$\frac{3 x^{2}}{x-1}+2=\frac{x}{x-1}$$

Solve each equation by the square root property. $$x^{2}=121$$

Solve each inequality. Give the solution set using interval notation. $$|2 x+5|<3$$

What is the length of the side of a square if its area and perimeter are numerically equal?

Solve each equation. $$0.08 x+0.06(x+12)=7.72$$

(This problem appears in the 1994 movie Little Big League.) If Joe can paint a house in \(3 \mathrm{hr},\) and Sam can paint the same house in \(5 \mathrm{hr},\) how long does it take them to do it together?

Solve each equation by the square root property. $$27-x^{2}=0$$

Wind Speed Joe traveled against the wind in a small plane for 3 hr. The return trip with the wind took 2.8 hr. Find the speed of the wind if the speed of the plane in still air is \(180 \mathrm{mph}\)

Solve each inequality. Give the solution set using interval notation. $$|3 x-4|<2$$

A rectangle has an area that is numerically twice its perimeter. If the length is twice the width, what are its dimensions?

Solve each equation. $$0.04(x-12)+0.06 x=1.52$$

Solve each equation by the square root property. $$48-x^{2}=0$$

Acid Mixture How many gallons of a \(5 \%\) acid solution must be mixed with 5 gal of a \(10 \%\) solution to obtain a \(7 \%\) solution?(TABLE CAN'T COPY)

Solve each inequality. Give the solution set using interval notation. $$|2 x+5| \geq 3$$

A can of Blue Runner Red Kidney Beans has surface area \(371 \mathrm{cm}^{2}\). Its height is \(12 \mathrm{cm} .\) What is the radius of the circular top? Round to the nearest hundredth.

Solve each inequality. Write each solution set in interval notation. $$-5<5+2 x<11$$

Solve each equation by the square root property. $$x^{2}=-81$$

Acid Mixture Marin Caswell needs \(10 \%\) hydrochloric acid for a chemistry experiment. How much \(5 \%\) acid should she mix with \(60 \mathrm{mL}\) of \(20 \%\) acid to get a \(10 \%\) solution?

Solve each inequality. Give the solution set using interval notation. $$|3 x-4| \geq 2$$

Determine whether each equation is an identity, a conditional equation, or \(a\) contradiction. Give the solution set. See Example 3 . $$\frac{1}{2}(6 x+20)=x+4+2(x+3)$$

A sewage treatment plant has two inlet pipes to its settling pond. One pipe can fill the pond 3 times as fast as the other pipe, and together they can fill the pond in 12 hr. How long will it take the faster pipe to fill the pond alone?

Solve each inequality. Write each solution set in interval notation. $$-7<2+3 x<5$$

Solve each equation by the square root property. $$x^{2}=-400$$

Alcohol Mixture Beau Glaser wishes to strengthen a mixture from \(10 \%\) alcohol to \(30 \%\) alcohol. How much pure alcohol should be added to \(7 \mathrm{L}\) of the \(10 \%\) mixture?

An inlet pipe can fill Blake's pool in \(5 \mathrm{hr}\), while an outlet pipe can empty it in 8 hr. In his haste to surf the Internet, Blake left both pipes open. How long did it take to fill the pool?

Solve each inequality. Write each solution set in interval notation. $$10 \leq 2 x+4 \leq 16$$

Solve each equation by the square root property. $$(3 x-1)^{2}=12$$

Solve each inequality. Give the solution set using interval notation. $$\left|\frac{3}{5}+x\right|<1$$

Saline Solution How much water should be added to \(8 \mathrm{mL}\) of \(6 \%\) saline solution to reduce the concentration to \(4 \% ?\)

With both taps open, Robert can fill his kitchen sink in 5 min. When full, the sink drains in 10 min. How long will it take to fill the sink if Robert forgets to put in the stopper?

Solve each inequality. Write each solution set in interval notation. $$-11>-3 x+1>-17$$

Solve each inequality. Give the solution set using interval notation. $$|5-3 x|>7$$

Solve each equation. $$x-\sqrt{2 x+3}=0$$

Solve each inequality. Write each solution set in interval notation. $$-4 \leq \frac{x+1}{2} \leq 5$$

Buying and Selling Land Roger bought two plots of land for a total of \(\$ 120,000\). When he sold the first plot, he made a profit of \(15 \% .\) When he sold the second, he lost \(10 \% .\) His total profit was \(\$ 5500 .\) How much did he pay for each piece of land?

Solve each inequality. Give the solution set using interval notation. $$|7-3 x|>4$$

Tanner Jones and Sheldon Furst have received communications receivers for Christmas. If they leave from the same point at the same time, Tanner walking north at \(2.5 \mathrm{mph}\) and Sheldon walking east at \(3 \mathrm{mph}\), how long will they be able to talk to each other if the range of the communications receivers is 4 mi? Round your answer to the nearest minute.

Solve each equation. $$x-\sqrt{3 x+18}=0$$

Solve each inequality. Write each solution set in interval notation. $$-5 \leq \frac{x-3}{3} \leq 1$$

Solve each equation by the square root property. $$(-2 x+5)^{2}=-8$$

Solve each inequality. Give the solution set using interval notation. $$|5-3 x| \leq 7$$