Chapter 1

Solve each equation in Exercises 73-98 by the method of your choice. \(3 x^{2}-12 x+12=0\)

The formula $$ S=4 \sqrt{x}+280 $$ models the average science test score, \(S, x\) years after 1982 . Use the formula to solve When will the average science score return to the 1970 average of \(304 ?\)

In a film, the actor Charles Coburn plays an elderly "uncle" character criticized for marrying a woman when he is 3 times her age. He wittily replies, "Ah, but in 20 years time I shall only be twice her age." How old is the "uncle" and the woman?

What is an inconsistent equation? Give an example.

Solve each equation in Exercises 73-98 by the method of your choice. \(9-6 x+x^{2}=0\)

The formula $$ S=4 \sqrt{x}+280 $$ models the average science test score, \(S, x\) years after 1982 . Use the formula to solve When will the average science test score be \(300 ?\) Out of a group of \(50,000\) births, the number of people, \(y\) surviving to age \(x\) is modeled by the formula $$ y=5000 \sqrt{100-x} $$

Suppose that we agree to pay you 8e for every problem in this chapter that you solve correctly and fine you \(5 \varnothing\) for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?

Solve each equation in Exercises 73-98 by the method of your choice. \(4 x^{2}-16=0\)

For Exercises \(86-89\) use your graphing utility to graph. each side of the equations in the same viewing rectangle. Based on the resulting graph, label each equation as conditional, inconsistent, or an identity. If the equation is conditional, use the \(x\) -coordinate of the intersection point to find the solution set. Verify this value by direct substitution into the equation. $$ 9 x+3-3 x=2(3 x+1) $$

Solve each equation in Exercises 73-98 by the method of your choice. \(3 x^{2}-27=0\)

Solve for \(C: \quad V=C-\frac{C-S}{L} N\)

For Exercises \(86-89\) use your graphing utility to graph. each side of the equations in the same viewing rectangle. Based on the resulting graph, label each equation as conditional, inconsistent, or an identity. If the equation is conditional, use the \(x\) -coordinate of the intersection point to find the solution set. Verify this value by direct substitution into the equation. $$ 2\left(x+\frac{1}{2}\right)=5 x+1-3 x $$

Solve each equation in Exercises 73-98 by the method of your choice. \(x^{2}-6 x+13=0\)

For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The formula $$ E=0.2 x^{3 / 2} $$ models the number of Earth days in a planet's year, \(E,\) where \(x\) is the average distance of the planet from the sun, in millions of kilometers Use the formula to solve We, of course, have 365 Earth days in our year. What is the average distance of Earth from the sun? Use a calculator and round to the nearest million kilometers.

One of the best ways to learn how to solve a word problem in algebra is to design word problems of your own. Creating a word problem makes you very aware of precisely how much information is needed to solve the problem. You must also focus on the best way to present information to a reader and on how much information to give. As you write your problem, you gain skills that will help you solve problems created by others. The group should design five different word problems that can be solved using linear equations. All of the problems should be on different topics. For example, the group should not have more than one problem on simple interest. The group should turn in both the problems and their algebraic solutions.

For Exercises \(86-89\) use your graphing utility to graph. each side of the equations in the same viewing rectangle. Based on the resulting graph, label each equation as conditional, inconsistent, or an identity. If the equation is conditional, use the \(x\) -coordinate of the intersection point to find the solution set. Verify this value by direct substitution into the equation. $$ \frac{2 x-1}{3}-\frac{x-5}{6}=\frac{x-3}{4} $$

Solve each equation in Exercises 73-98 by the method of your choice. \(x^{2}-4 x+29=0\)

For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The formula $$ E=0.2 x^{3 / 2} $$ models the number of Earth days in a planet's year, \(E,\) where \(x\) is the average distance of the planet from the sun, in millions of kilometers Use the formula to solve There are approximately 88 Earth days in the year of the planet Mercury. What is the average distance of Mercury from the sun? Use a calculator and round to the nearest million kilometers.

Which one of the following is true? a. The equation \(-7 x=x\) has no solution. b. The equations \(\frac{x}{x-4}=\frac{4}{x-4}\) and \(x=4\) are equivalent. c. The equations \(3 y-1=11\) and \(3 y-7=5\) are equivalent. d. If \(a\) and \(b\) are any real numbers, then \(a x+b=0\) always has one number in its solution set.

Solve each equation in Exercises 73-98 by the method of your choice. \(x^{2}=4 x-7\)

Solve for \(x: \quad a x+b=c\)

Solve each equation in Exercises 73-98 by the method of your choice. \(5 x^{2}=2 x-3\)

Write three equations that are equivalent to \(x=5\)

Solve each equation in Exercises 73-98 by the method of your choice. \(2 x^{2}-7 x=0\)

Without actually solving the equation, give a general description of how to solve \(x^{3}-5 x^{2}-x+5=0\)

The line graph at the top of the next column shows the declining consumption of cigarettes in the United States. The data shown by the graph can be modeled by $$N=550-9 x$$ where \(N\) is the number of cigarettes consumed, in billions, \(x\) years after \(1988 .\) Use this formula to solve Exercises 93-94. (Line graph cannot copy) How many years after 1988 will cigarette consumption be less than 370 billion cigarettes each year? Which years does this describe?

If \(x\) represents a number, write an English sentence about the number that results in an inconsistent equation.

Solve each equation in Exercises 73-98 by the method of your choice. \(2 x^{2}+5 x=3\)

In solving \(\sqrt{3 x+4}-\sqrt{2 x+4}=2,\) why is it a good idea to isolate a radical term? What if we don't do this and simply square each side? Describe what happens.

The line graph at the top of the next column shows the declining consumption of cigarettes in the United States. The data shown by the graph can be modeled by $$N=550-9 x$$ where \(N\) is the number of cigarettes consumed, in billions, \(x\) years after \(1988 .\) Use this formula to solve. (Line graph cannot copy) Describe how many years after 1988 cigarette consumption will be less than 325 billion cigarettes each year. Which years are included in your description?

Find \(b\) such that \(\frac{7 x+4}{b}+13=x\) will have a solution set given by \(\\{-6\\}\)

Solve each equation in Exercises 73-98 by the method of your choice. \(\frac{1}{x}+\frac{1}{x+2}=\frac{1}{3}\)

The formula for converting Fahrenheit temperature, \(F\) to Celsius temperature, \(C\), is $$C=\frac{5}{9}(F-32)$$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

Find \(b\) such that \(\frac{4 x-b}{x-5}=3\) will have a solution set given by \(\varnothing\)

Solve each equation in Exercises 73-98 by the method of your choice. \(\frac{1}{x}+\frac{1}{x+3}=\frac{1}{4}\)

Explain how to recognize an equation that is quadratic in form. Provide two original examples with your explanation.

The formula $$T=0.01 x+56.7$$ models the global mean temperature, \(T,\) in degrees Fahrenheit, of Earth \(x\) years after \(1905 .\) For which range of years was the global mean temperature at least \(56.7^{\circ} \mathrm{F}\) and at most \(57.2^{\circ} \mathrm{F} ?\)

In your group, describe the best procedure for solving the following equation: $$0.47 x+\frac{19}{4}=-0.2+\frac{2}{5} x$$ Use this procedure to actually solve the equation. Then compare procedures with other groups working on this problem. Which group devised the most streamlined method?

Solve each equation in Exercises 73-98 by the method of your choice. \(\frac{2 x}{x-3}+\frac{6}{x+3}=-\frac{28}{x^{2}-9}\)

Describe two methods for solving this equation: \(x-5 \sqrt{x}+4=0\)

The three television programs viewed by the greatest percentage of U.S. households in the twentieth century are shown in the table. The data are from a random survey of 4000 TV households by Nielsen Media Research. In Exercises 97-98, let \(x\) represent the actual viewing percentage in the U.S. population. TV Programs with the Greatest U.S. Audience Viewing Percentage of the Twentieth Century $$\begin{array}{|c|c|} \hline \text { Program } & \begin{array}{l} \text { Viewing } \\ \text { Percentage in } \\ \text { Survey } \end{array} \\ \hline \begin{array}{l} 1 .^{*} \mathrm{M}^{*} \mathrm{A}^{*} \mathrm{S}^{*} \mathrm{H}^{\prime \prime} \\ \text { Feb. } 28,1983 \end{array} & 60.2 \% \\ \hline \begin{array}{l} \text { 2. "Dallas" } \\ \text { Nov. } 21,1980 \end{array} & 53.3 \% \\ \hline \begin{array}{l} \text { 3. "Roots" Part } 8 \\ \text { Jan. } 30,1977 \end{array} & 51.1 \% \\ \hline \end{array}$$ The inequality \(|x-60.2| \leq 1.6\) describes the actual viewing percentage for "M*A*S*H" in the U.S. population. Solve the inequality and interpret the solution. Explain why the surveys margin of error is \(\pm 1.6 \%\)

Solve each equation in Exercises 73-98 by the method of your choice. $$\frac{3}{x-3}+\frac{5}{x-4}=\frac{x^{2}-20}{x^{2}-7 x+12}$$

The three television programs viewed by the greatest percentage of U.S. households in the twentieth century are shown in the table. The data are from a random survey of 4000 TV households by Nielsen Media Research. Let \(x\) represent the actual viewing percentage in the U.S. population. TV Programs with the Greatest U.S. Audience Viewing Percentage of the Twentieth Century $$\begin{array}{|c|c|} \hline \text { Program } & \begin{array}{l} \text { Viewing } \\ \text { Percentage in } \\ \text { Survey } \end{array} \\ \hline \begin{array}{l} 1 .^{*} \mathrm{M}^{*} \mathrm{A}^{*} \mathrm{S}^{*} \mathrm{H}^{\prime \prime} \\ \text { Feb. } 28,1983 \end{array} & 60.2 \% \\ \hline \begin{array}{l} \text { 2. "Dallas" } \\ \text { Nov. } 21,1980 \end{array} & 53.3 \% \\ \hline \begin{array}{l} \text { 3. "Roots" Part } 8 \\ \text { Jan. } 30,1977 \end{array} & 51.1 \% \\ \hline \end{array}$$ The inequality \(|x-51.1| \leq 1.6\) describes the actual viewing percentage for "Roots" Part 8 in the U.S. population. Solve the inequality and interpret the solution. Explain why the surveys margin of error is \(\pm 1.6 \%\)

If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if \(h,\) the number of outcomes that result in heads, satisfies \(\left|\frac{h-50}{5}\right| \geq 1.645\) Describe the number of outcomes that determine an unfair coin that is tossed 100 times.

The inequality \(|T-57| \leq 7\) describes the range of monthly average temperature, \(T,\) in degrees Fahrenheit, for San Francisco, California. The inequality \(|T-50| \leq 22\) describes the range of monthly average temperature, \(T\), in degrees Fahrenheit, for Albany, New York. Solve each inequality and interpret the solution. Then describe at least three differences between the monthly average temperatures for the two cities.

The Food Stamp Program is America's first line of defense against hunger for millions of families. Over half of all participants are children; one out of six is a low-income older adult. Exercises \(101-104\) involve the number of participants in the program from 1990 through 2000 . The formula $$ y=-\frac{1}{2} x^{2}+4 x+19 $$ models the number of people, \(y,\) in millions, receiving food stamps \(x\) years after \(1990 .\) Use the formula to solve Exercises 101-102 In which year did 27 million people receive food stamps?

use a graphing utility and the graph's \(x\)-intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$ \begin{aligned} &x^{3}+3 x^{2}-x-3=0\\\ &[-6,6,1] \text { by }[-6,6,1] \end{aligned} $$

In Exercises 101-110, use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. A truck can be rented from Basic Rental for \(\$ 50\) a day plus \(\$ 0.20\) per mile. Continental charges \(\$ 20\) per day plus \(\$ 0.50\) per mile to rent the same truck. How many miles must be driven in a day to make the rental cost for Basic Rental a better deal then Continental's?

The Food Stamp Program is America's first line of defense against hunger for millions of families. Over half of all participants are children; one out of six is a low-income older adult. Exercises \(101-104\) involve the number of participants in the program from 1990 through 2000 . The formula $$ y=-\frac{1}{2} x^{2}+4 x+19 $$ models the number of people, \(y,\) in millions, receiving food stamps \(x\) years after \(1990 .\) Use the formula to solve Exercises 101-102 In which years did 19 million people receive food stamps?

use a graphing utility and the graph's \(x\)-intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$ \begin{aligned} &-x^{4}+4 x^{3}-4 x^{2}=0\\\ &[-6,6,1] \text { by }[-9,2,1] \end{aligned} $$