Chapter 1

Solve each equation in by making an appropriate substitution. $$ 2 x^{2 / 3}+7 x^{1 / 3}-15=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}-3 x-5=0 $$

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$6

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{1}{x-4}-\frac{5}{x+2}=\frac{6}{x^{2}-2 x-8} $$

A stand-up comedian uses algebra in some jokes, Fincluding one about a telephone recording that eannounces "You have just reached an imaginary number. Please multiply by \(i\) and dial again." Explain the joke.

Use the position formula $$ s=-16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{0}=\text { initial velocity, } s_{0}=\text { initial position, } t=\text { time }\right)\) to answer Exercises \(49-52 .\) If necessary, round answers to the nearest hundredth of a second. A projectile is fired straight upward from ground level with an initial velocity of 128 feet per second. During which interval of time will the projectile's height exceed 128 feet?

Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.

Solve each equation in by making an appropriate substitution. $$ x^{2 / 5}+x^{1 / 5}-6=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ 2 x^{2}-7 x+3=0 $$

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$7

Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{6}{x+3}-\frac{5}{x-2}=\frac{-20}{x^{2}+x-6} $$

Use the position formula $$ s=-16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{0}=\text { initial velocity, } s_{0}=\text { initial position, } t=\text { time }\right)\) to answer Exercises \(49-52 .\) If necessary, round answers to the nearest hundredth of a second. A ball is thrown vertically upward with a velocity of 64 feet per second from the top edge of a building 80 feet high. For how long is the ball higher than 96 feet?

Explain why \((5,-2)\) and \((-2,5)\) do not represent the same point.

Solve each equation in by making an appropriate substitution. $$ 2 x-3 x^{1 / 2}+1=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ 4 x^{2}-4 x-1=0 $$

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$-3 \leq x-2<1$$

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ 4(x-7)=4 x-28 $$

Explain the error in Exercises \(51-52\) \((\sqrt{-9})^{2}=\sqrt{-9} \cdot \sqrt{-9}=\sqrt{81}=9\)

Use the position formula $$ s=-16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{0}=\text { initial velocity, } s_{0}=\text { initial position, } t=\text { time }\right)\) to answer Exercises \(49-52 .\) If necessary, round answers to the nearest hundredth of a second. A diver leaps into the air at 20 feet per second from a diving board that is 10 feet above the water. For how many scconds is the diver at least 12 feet above the water?

Explain how to graph an equation in the rectangular coordinate system.

Solve each equation in by making an appropriate substitution. $$ x+3 x^{1 / 2}-4=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ 2 x^{2}-4 x-1=0 $$

After a graphing calculator's price is reduced by \(\frac{1}{3}\) of its original price, you purchase it for \(\$ 64 .\) What was the graphing calculator's price before the reduction?

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$-6

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ 4(x-7)=4 x+28 $$

Which one of the following is true? a. Some irrational numbers are not complex numbers. b. \((3+7 i)(3-7 i)\) is an imaginary number. c. \(\frac{7+3 i}{5+3 i}=\frac{7}{5}\) d. In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) can be factored as \((x+y i)(x-y i)\)

What does a \([-20,2,1]\) by \([-4,5,0.5]\) viewing rectangle mean?

Solve each equation in by making an appropriate substitution. $$ (x-5)^{2}-4(x-5)-21=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ 3 x^{2}-2 x-2=0 $$

After a \(12 \%\) price reduction, a car sold for $$ 17,600 .$ What was the car's price before the reduction?

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$-11<2 x-1 \leq-5$$

In Exercises \(54-56,\) perform the indicated operations and write the result in standard form. \((8+9 i)(2-i)-(1-i)(1+i)\)

Solve each equation in by making an appropriate substitution. $$ (x-5)^{2}-4(x-5)-21=0 $$

Solve each equation in Exercises \(39-54\) by completing the square. $$ 3 x^{2}-5 x-10=0 $$

Including 8% sales tax, an inn charges $$ 162$ per night. Find the inn's nightly cost before the tax is added.

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$3 \leq 4 x-3<19$$

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ \frac{7 x}{x}=7 $$

In Exercises \(54-56,\) perform the indicated operations and write the result in standard form. $$\frac{4}{(2+i)(3-i)}$$

Solve each equation in by making an appropriate substitution. $$ \left(x^{2}-x\right)^{2}-14\left(x^{2}-x\right)+24=0 $$

Solve each equation in Exercises \(55-64\) using the quadratic formula. $$ x^{2}+8 x+15=0 $$

An HMO pamphlet contains the following recommended weight for women: "Give yourself 100 pounds for the first 5 feet plus 5 pounds for every inch over 5 feet tall." Using this description, what height corresponds to a recommended weight of 135 pounds?

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$-3 \leq \frac{2}{3} x-5<-1$$

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ 4 x+5 x=8 x $$

In Exercises \(54-56,\) perform the indicated operations and write the result in standard form. $$\frac{1+i}{1+2 i}+\frac{1-i}{1-2 i}$$

The stated intent of the 1994 "don't ask, don't tell" policy was to reduce the number of discharges of gay men and lesbians from the military. The equation $$y=45.48 x^{2}-334.35 x+1237.9$$ describes the number of gay service members, \(y\) discharged from the military for homosexuality \(x\) years after \(1990 .\) Graph the equation in a \([0,10,1]\) by \([0,2200,200]\) viewing rectangle. Then describe something about the relationship between \(x\) and \(y\) that is revealed by looking at the graph that is not obvious from the equation. What does the graph reveal about the success or lack of success of "don't ask, don't tell"?

Solve each equation in by making an appropriate substitution. $$ \left(x^{2}-2 x\right)^{2}-11\left(x^{2}-2 x\right)+24=0 $$

Solve each equation in Exercises \(55-64\) using the quadratic formula. $$ x^{2}+8 x+12=0 $$

A job pays an annual salary of \(\$ 33,150\), which includes a holiday bonus of \(\$ 750 .\) If paychecks are issued twice a month, what is the gross amount for each paycheck?

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$-6 \leq \frac{1}{2} x-4<-3$$

In Exercises \(51-58,\) determine whether each equation is an identity, a conditional equation, or an inconsistent equation. $$ 8 x+2 x=9 x $$