Chapter 1

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

Mixture Problem What quantity of a 60\(\%\) acid solution must be mixed with a 30\(\%\) solution to produce 300 \(\mathrm{mL}\) of a 50\(\%\) solution?

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

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ 0<|x-5| \leq \frac{1}{2} $$

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

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

\(5-60\) Find all real solutions of the equation. $$ \sqrt{x}-3 \sqrt[4]{x}-4=0 $$

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

Mixture Problem What quantity of pure acid must be added to 300 \(\mathrm{mL}\) of a 50\(\%\) acid solution to produce a 60\(\%\) acid solution?

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

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ \frac{1}{|x+7|}>2 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2}<4 $$

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

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

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

Mixture Problem A jeweler has five rings, each weighing \(18 \mathrm{g},\) made of an alloy of 10\(\%\) silver and 90\(\%\) gold. She decides to melt down the rings and add enough silver to reduce the gold content to \(75 \% .\) How much silver should she add?

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

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ \frac{1}{|2 x-3|} \leq 5 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{2} \geq 9 $$

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

\(5-60\) Find all real solutions of the equation. $$ 2(x-4)^{7 / 3}-(x-4)^{4 / 3}-(x-4)^{1 / 3}=0 $$

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

Mixture Problem A pot contains 6 \(\mathrm{L}\) of brine at a concentration of 120 \(\mathrm{g} / \mathrm{L}\) . How much of the water should be boiled off to increase the concentration to 200 \(\mathrm{g} / \mathrm{L}\) ?

Evaluate the radical expression, and express the result in the form \(a+b i\) $$ \sqrt{-3} \sqrt{-12} $$

\(49-52=\) A phrase that describes a set of real numbers is given. Express the phrase as an inequality involving an absolute value. All real numbers \(x\) less than 3 units from 0

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

Find all real solutions of the equation. $$ 10 y^{2}-16 y+5=0 $$

\(5-60\) Find all real solutions of the equation. $$ x^{3 / 2}+8 x^{1 / 2}+16 x^{-1 / 2}=0 $$

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(\frac{3}{x+4}=\frac{1}{x}+\frac{6 x+12}{x^{2}+4 x}\)

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 radical expression, and express the result in the form \(a+b i\) $$ \sqrt{4} \sqrt{-27} $$

\(49-52=\) A phrase that describes a set of real numbers is given. Express the phrase as an inequality involving an absolute value. All real numbers \(x\) more than 2 units from 0

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ (x-5)(x-2)(x+1)>0 $$

Find all real solutions of the equation. $$ 25 x^{2}+70 x+49=0 $$

\(5-60\) Find all real solutions of the equation. $$ x^{1 / 2}+3 x^{-1 / 2}=10 x^{-3 / 2} $$

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

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 radical expression, and express the result in the form \(a+b i\) $$ (3-\sqrt{-5})(1+\sqrt{-1}) $$

\(49-52=\) A phrase that describes a set of real numbers is given. Express the phrase as an inequality involving an absolute value. All real numbers \(x\) at least 5 units from 7

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ (x-4)(x+2)^{2}<0 $$

Find all real solutions of the equation. $$ 3 x^{2}+2 x+2=0 $$

\(5-60\) Find all real solutions of the equation. $$ X^{1 / 2}-3 X^{1 / 3}=3 X^{1 / 6}-9 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}=49\)

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 radical expression, and express the result in the form \(a+b i\) $$ (\sqrt{3}-\sqrt{-4})(\sqrt{6}-\sqrt{-8}) $$

\(49-52=\) A phrase that describes a set of real numbers is given. Express the phrase as an inequality involving an absolute value. All real numbers \(x\) at most 4 units from 2

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

Find all real solutions of the equation. $$ 5 x^{2}-7 x+5=0 $$

\(5-60\) Find all real solutions of the equation. $$ x-5 \sqrt{x}+6=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}=18\)