Chapter 1

Mixture Problem The radiator in a car is filled with a solution of 60\(\%\) antifreeze and 40\(\%\) water. The manufacturer of the antifreeze suggests that, for summer driving, optimal cooling of the engine is obtained with only 50\(\%\) antifreeze. If the capacity of the radiator is 3.6 \(\mathrm{L}\) , how much coolant should be drained and replaced with water to reduce the antifreeze concentration to the recommended level?

Evaluate the expression and write the result in the form \(a+b i .\) $$ \frac{1}{1+i}-\frac{1}{1-i} $$

1–54 ? Find all real solutions of the equation. $$ x^{3 / 2}+8 x^{1 / 2}+16 x^{-1 / 2}=0 $$

Find all real solutions of the equation. \(2 y^{2}-y-\frac{1}{2}=0\)

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ x-\frac{1}{3} x-\frac{1}{2} x-5=0 $$

Solve the inequality. Express the answer using interval notation. $$ 7|x+2|+5>4 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}+5 x+6>0 $$

Mixture Problem A health clinic uses a solution of bleach to sterilize petri dishes in which cultures are grown. The sterilization tank contains 100 gal of a solution of 2\(\%\) ordinary household bleach mixed with pure distilled water. New research indicates that the concentration of bleach should be 5\(\%\) for complete sterilization. How much of the solution should be drained and replaced with bleach to increase the bleach content to the recommended level?

Evaluate the expression and write the result in the form \(a+b i .\) $$ \frac{(1+2 i)(3-i)}{2+i} $$

1–54 ? Find all real solutions of the equation. $$ x^{1 / 2}+3 x^{-1 / 2}=10 x^{-3 / 2} $$

Find all real solutions of the equation. \(\theta^{2}-\frac{3}{2} \theta+\frac{9}{16}=0\)

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ 2 x-\frac{x}{2}+\frac{x+1}{4}=6 x $$

Solve the inequality. Express the answer using interval notation. $$ \frac{1}{2}\left|4 x+\frac{1}{3}\right|>\frac{5}{6} $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ 2 x^{2}+x \geq 1 $$

Mixture Problem \(A\) bottle contains 750 \(\mathrm{mL}\) of fruit punch with a concentration of 50\(\%\) pure fruit juice. Jill drinks 100 \(\mathrm{mL}\) of the punch and then refills the bottle with an equal amount of a cheaper brand of punch. If the concentration of juice in the bottle is now reduced to \(48 \%,\) what was the concentration in the punch that Jill added?

Evaluate the expression and write the result in the form \(a+b i .\) $$ i^{3} $$

1–54 ? Find all real solutions of the equation. $$ x^{1 / 2}-3 x^{1 / 3}=3 x^{1 / 6}-9 $$

Find all real solutions of the equation. \(4 x^{2}+16 x-9=0\)

Solve the inequality. Express the answer using interval notation. $$ 2\left|\frac{1}{2} x+3\right|+3 \leq 51 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}

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 expression and write the result in the form \(a+b i .\) $$ (2 i)^{4} $$

1–54 ? Find all real solutions of the equation. $$ x-5 \sqrt{x}+6=0 $$

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

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{1}{1-\frac{3}{2+w}}=60 $$

Solve the inequality. Express the answer using interval notation. $$ 1 \leq|x| \leq 4 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ 3 x^{2}-3 x<2 x^{2}+4 $$

Length of a Shadow \(A\) man is walking away from a lamppost with a light source 6 \(\mathrm{m}\) above the ground. The man is 2 \(\mathrm{m}\) tall. How long is the man's shadow when he is 10 \(\mathrm{m}\) from the lamppost? [Hint: Use similar triangles.]

Evaluate the expression and write the result in the form \(a+b i .\) $$ i^{100} $$

1–54 ? Find all real solutions of the equation. $$ \frac{1}{x^{3}}+\frac{4}{x^{2}}+\frac{4}{x}=0 $$

Find all real solutions of the equation. \(3+5 z+z^{2}=0\)

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{u}{u-\frac{u+1}{2}}=4 $$

Solve the inequality. Express the answer using interval notation. $$ 0<|x-5| \leq \frac{1}{2} $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ 5 x^{2}+3 x \geq 3 x^{2}+2 $$

Evaluate the expression and write the result in the form \(a+b i .\) $$ i^{1002} $$

1–54 ? Find all real solutions of the equation. $$ 4 x^{-4}-16 x^{-2}+4=0 $$

Find all real solutions of the equation. \(w^{2}=3(w-1)\)

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{1}{3-t}+\frac{4}{3+t}+\frac{16}{9-t^{2}}=0 $$

Solve the inequality. Express the answer using interval notation. $$ \frac{1}{|x+7|}>2 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}>3(x+6) $$

Law of the Lever The figure shows a lever system, similar to a seesaw that you might find in a children's play-ground. For the system to balance, the product of the weight and its distance from the fulcrum must be the same on each side; that is $$ w_{1} x_{1}=w_{2} x_{2} $$ This equation is called the law of the lever, and was first discovered by Archimedes (see page 796 ). A woman and her son are playing on a seesaw. The boy is at one end, 8 ft from the fulcrum. If the son weighs 100 lb and the mother weighs 125 lb, where should the woman sit so that the seesaw is balanced?

Evaluate the expression and write the result in the form \(a+b i .\) $$ \sqrt{-25} $$

1–54 ? Find all real solutions of the equation. $$ \sqrt{2 x+1}+1=x $$

Find all real solutions of the equation. \(x^{2}-\sqrt{5} x+1=0\)

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ \frac{x}{2 x-4}-2=\frac{1}{x-2} $$

Solve the inequality. Express the answer using interval notation. $$ \frac{1}{|2 x-3|} \leq 5 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}+2 x>3 $$

Evaluate the expression and write the result in the form \(a+b i .\) $$ \sqrt{\frac{-9}{4}} $$

1–54 ? Find all real solutions of the equation. $$ x-\sqrt{9-3 x}=0 $$

Find all real solutions of the equation. \(\sqrt{6} x^{2}+2 x-\sqrt{\frac{3}{2}}=0\)