Chapter 2

A college charters a bus for \(\$ 1700\) to take a group of students to see a Broadway production. When 6 more students join the trip, the cost per student decreases by \(\$ 7.50 .\) How many students were in the original group?

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=-x^{2}+2 x+3\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 3\)

On the first part of a 280 -mile trip, a salesperson averaged 63 miles per hour. The salesperson averaged only 54 miles per hour on the last part of the trip because of an increased volume of traffic. (a) Write the total time \(t\) for the trip as a function of the distance \(x\) traveled at an average speed of 63 miles per hour. (b) Use a graphing utility to graph the time function. What is the domain of the function? (c) Approximate the number of miles traveled at 63 miles per hour when the total time is 4 hours and 45 minutes.

Write a linear equation that has the given solution. (There are many correct answers.) $$x=-3$$

Determine whether the statement is true or false. Justify your answer. The conjugate of the sum of two complex numbers is equal to the sum of the conjugates of the two complex numbers.

Three students are planning to rent an apartment for a year and share equally in the cost. By adding a fourth person, each person saves \(\$ 75\) a month. How much is the monthly rent?

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=x^{3}-x^{2}-16 x+16\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 36\)

A 55 -gallon barrel contains a mixture with a concentration of \(33 \%\) sodium chloride. You remove \(x\) gallons of this mixture and replace it with \(100 \%\) sodium chloride. (a) Write the amount \(A\) of sodium chloride in the final mixture as a function of \(x\) (b) Use a graphing utility to graph the concentration function. What is the domain of the function? (c) Approximate (accurate to one decimal place) the value of \(x\) when the final mixture is \(60 \%\) sodium chloride.

The coordinate system shown below is called the complex plane. In the complex plane, the point that corresponds to the complex number \(a+b i\) is \((a, b) .\) Match each complex number with its corresponding point. (i) \(2 \quad\) (ii) \(2 i \quad\) (iii) \(-2+i \quad\) (iv) \(1-2 i\)

Write a linear equation that has the given solution. (There are many correct answers.) $$x=0$$

Finance A deposit of \(\$ 7500\) in a mutual fund reaches a balance of \(\$ 11,752.45\) after 10 years. What annual interest rate on a certificate of deposit compounded monthly would yield an equivalent return?

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$\frac{1}{x}-x>0$$

Find two quadratic equations having the given solutions. (There are many correct answers.) $$-6,5$$

Write a linear equation that has the given solution. (There are many correct answers.) $$x=\frac{1}{4}$$

Twenty years ago, your parents deposited \(\$ 3000\) in a long-term investment in which interest was compounded biannually. Today, the value of the investment is \(\$ 8055.19 .\) What is the annual interest rate for this investment?

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$\frac{1}{x}-4<0$$

Find two quadratic equations having the given solutions. (There are many correct answers.) $$-2,1$$

Consider the binomials \(x+5\) and \(2 x-1\) and the complex numbers \(1+5 i\) and \(2-i\) (a) Find the sum of the binomials and the sum of the complex numbers. Describe the similarities and differences in your results. (b) Find the product of the binomials and the product of the complex numbers. Describe the similarities and differences in your results. (c) Explain why the products in part (b) are not related in the same way as the sums in part (a).

Write a linear equation that has the given solution. (There are many correct answers.) $$x=-2.5$$

The temperature \(T\) (in degrees Fahrenheit) of saturated steam increases as pressure increases. This relationship is approximated by the model $$T=75.82-2.11 x+43.51 \sqrt{x}, \quad 5 \leq x \leq 40$$ where \(x\) is the absolute pressure (in pounds per square inch). Approximate the pressure for saturated steam at a temperature of \(212^{\circ} \mathrm{F}.\)

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$\frac{x+6}{x+1}-2 \leq 0$$

Find two quadratic equations having the given solutions. (There are many correct answers.) $$-\frac{7}{3}, \frac{6}{7}$$

The following information describes a possible negative income tax for a family consisting of two adults and two children. The plan would guarantee the poor a minimum income while encouraging a family to increase its private income \((0 \leq x \leq 20,000)\) (A subsidy is a grant of money.) Family's earned income: \(I=x\) Subsidy: \(S=10,000-\frac{1}{2} x\) Total income: \(T=I+S\) (a) Write the total income \(T\) in terms of \(x\) (b) Use a graphing utility to find the earned income \(x\) when the subsidy is \(\$ 6600 .\) Verify your answer algebraically. (c) Use the graphing utility to find the earned income \(x\) when the total income is \(\$ 13,800 .\) Verify your answer algebraically. (d) Find the subsidy \(S\) graphically when the total income is \(\$ 12,500\)

Perform the operation and write the result in standard form. $$(4 x-5)(4 x+5)$$

(p. 200) An airline offers daily flights between Chicago and Denver. The total monthly cost \(C\) (in millions of dollars) of these flights is modeled by \(c=\sqrt{0.2 x+1}\) where \(x\) is the number of passengers (in thousands). The total cost of the flights for June is 2.5 million dollars. How many passengers flew in June?

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$\frac{x+12}{x+2}-3 \geq 0$$

Find two quadratic equations having the given solutions. (There are many correct answers.) $$-\frac{2}{3}, \frac{4}{3}$$

The median weekly earnings \(y\) (in dollars) of full-time workers in the United States from 2000 through 2012 can be modeled by \(y=16.9 t+574,0 \leq t \leq 12\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (a) Find algebraically and interpret the \(y\) -intercept of the model. (b) What is the slope of the model and what does it tell you about the median weekly earnings of full-time workers in the United States? (c) Do you think the model can be used to predict the median weekly earnings of full-time workers in the United States for years beyond \(2012 ?\) If so, for what time period? Explain. (d) Explain, both algebraically and graphically, how you could find when the median weekly earnings of full-time workers reaches \(\$ 800 .\)

Perform the operation and write the result in standard form. $$(x+2)^{3}$$

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=\frac{3 x}{x-2}\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 6\)

Find two quadratic equations having the given solutions. (There are many correct answers.) $$5 \sqrt{3},-5 \sqrt{3}$$

The populations (in thousands) of Maryland \(M\) and Wisconsin \(W\) from 2001 through 2013 can be modeled by \(M=43.4 t+5355,1 \leq t \leq 13\) \(W=28.4 t+5398,1 \leq t \leq 13\) where \(t\) represents the year, with \(t=1\) corresponding to \(2001 .\) (a) Use a graphing utility to graph each model in the same viewing window over the appropriate domain. Approximate the point of intersection. Round your result to one decimal place. Explain the meaning of the coordinates of the point. (b) Find the point of intersection algebraically. Round your result to one decimal place. What does the point of intersection represent? (c) Explain the meaning of the slopes of both models and what they tell you about the population growth rates. (d) Use the models to estimate the population of each state in \(2016 .\) Do the values seem reasonable? Explain.

Perform the operation and write the result in standard form. $$\left(3 x-\frac{1}{2}\right)(x+4)$$

Consider the equation $$\frac{6}{(x-3)(x-1)}=\frac{3}{x-3}+\frac{4}{x-1}$$ Without performing any calculations, explain how to clear this equation of fractions. Is it possible that this process will introduce an extraneous solution? If so, describe two ways to determine whether a solution is extraneous.

Meteorology A meteorologist is positioned 100 feet from the point at which a weather balloon is launched. When the balloon is at height \(h,\) the distance \(d\) (in feet) between the meteorologist and the balloon is given by \(d=\sqrt{100^{2}+h^{2}}\) (a) Use a graphing utility to graph the equation. Use the trace feature to approximate the value of \(h\) when \(d=200.\) (b) Complete the table. Use the table to approximate the value of \(h\) when \(d=200.\) $$\begin{array}{|c|c|c|c|c|c|c|} \hline h & 160 & 165 & 170 & 175 & 180 & 185 \\ \hline d & & & & & & \\ \hline \end{array}$$ (c) Find \(h\) algebraically when \(d=200.\) (d) Compare the results of each method. In each case, what information did you gain that wasn't revealed by another solution method?

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=\frac{5 x}{x^{2}+4}\) Inequalities (a) \(y \geq 1\) (b) \(y \leq 0\)

Find two quadratic equations having the given solutions. (There are many correct answers.). $$2 \sqrt{5},-2 \sqrt{5}$$

Perform the operation and write the result in standard form. $$(2 x-5)^{2}$$

Find \(c\) such that \(x=2\) is a solution of the linear equation \(5 x+2 c=12+4 x-2 c\)

The numbers of crimes (in millions) committed in the United States from 2008 through 2012 can be approximated by the model \(C=\sqrt{1.49145 t^{2}-35.034 t+309.6}, 8 \leq t \leq 12\) where \(t\) is the year, with \(t=8\) corresponding to 2008 (Source: Federal Bureau of Investigation) (a) Use the table feature of a graphing utility to estimate the number of crimes committed in the U.S. each year from 2008 through 2012 . (b) According to the table, when was the first year that the number of crimes committed fell below 11 million? (c) Find the answer to part (b) algebraically. (d) Use the graphing utility to graph the model and find the answer to part (b).

Find the domain of \(x\) in the expression. $$\sqrt{x-5}$$

Find two quadratic equations having the given solutions. (There are many correct answers.) $$1+2 \sqrt{3}, 1-2 \sqrt{3}$$

Determine whether the statement is true or false. Justify your answer. To find the \(y\) -intercept of a graph, let \(x=0\) and solve the equation for \(y\).

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=5-4 x$$

Find the domain of \(x\) in the expression. $$\sqrt{6 x+15}$$

Find two quadratic equations having the given solutions. (There are many correct answers.) $$2+3 \sqrt{5}, 2-3 \sqrt{5}$$

Determine whether the statement is true or false. Justify your answer. Every linear equation has at least one \(y\) -intercept or \(x\) -intercept.

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=\frac{3 x-5}{2}+2$$

Find the domain of \(x\) in the expression. $$\sqrt{-x^{2}+x+12}$$

You graphically approximate the solution of the equation \(\frac{x}{x-1}-\frac{99}{100}=0\) to be \(x=-99.1 .\) Substituting this value for \(x\) produces \(\frac{-99.1}{-99.1-1}-\frac{99}{100}=0.00000999=9.99 \times 10^{-6}\) Is -99.1 a good approximation of the solution? Write a short paragraph explaining why or why not.