Chapter 1

Investment problems such as those in Exercises \(75-80\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\) where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Retirement Planning In planning her retirement, Mary Lynn Ellis deposits some money at \(2.5 \%\) interest with twice as much deposited at \(3 \% .\) Find the amount deposited at each rate if the total annual interest income is \(\$ 850\).

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(3 x-6=0\) (b) \(3 x-6>0\) (c) \(3 x-6<0\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$a=8, b=15 ; \text { find } c$$

Sketch by hand the graph of the line passing through the given point and having the given slope. Label Through \((-2,-3), m=-\frac{3}{4}\)

Investment problems such as those in Exercises \(75-80\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\) where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Investing a Building Fund A church building fund has invested some money in two ways: part of the money at \(4 \%\) interest and four times as much at \(3.5 \% .\) Find the amount invested at each rate if the total annual income from interest is \(\$ 3600\).

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(5 x+10=0\) (b) \(5 x+10>0\) (c) \(5 x+10<0\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$a=7, b=24 ; \text { find } c$$

Explain each term in your own words. (a) Relation (b) Function (c) Domain of a function (d) Range of a function (e) Independent variable (1) Dependent variable

Sketch by hand the graph of the line passing through the given point and having the given slope. Label Through \((-2,-3), m=-\frac{3}{4}\)

Investment problems such as those in Exercises \(75-80\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\) where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Lottery Winnings Nancy B. Kindy won \(\$ 200,000\) in a state lottery. She first paid income tax of \(30 \%\) on the winnings. Of the rest, she invested some at \(1.5 \%\) and some at \(4 \%,\) earning \(\$ 4350\) interest per year. How much did she invest at each rate?

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(1-2 x=0\) (b) \(1-2 x \leq 0\) (c) \(1-2 x \geq 0\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$a=13, c=85 ; \text { find } b$$

Sketch by hand the graph of the line passing through the given point and having the given slope. Label Through \((-2,-3), m=-\frac{3}{4}\)

Investment problems such as those in Exercises \(75-80\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\) where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Cookbook Royalties Latasha Williams earned \(\$ 48,000\) from royalties on her cookbook. She paid a \(28 \%\) income tax on these royalties. The balance was invested in two ways, some of it at \(3.25 \%\) interest and some at \(1.75 \% .\) The investments produced \(\$ 904.80\) interest the first year. Find the amount invested at each rate.

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(4-3 x=0\) (b) \(4-3 x \leq 0\) (c) \(4-3 x \geq 0\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$a=14, c=50 ; \text { find } b$$

Sketch by hand the graph of the line passing through the given point and having the given slope. Label Through \((-2,-3), m=-\frac{3}{4}\)

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(x+12=4 x\) (b) \(x+12>4 x\) (c) \(x+12<4 x\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$a=5, b=8 ; \text { find } c$$

The following table lists Square's daily transactions \(y\) in millions of dollars, \(x\) months after March 2011 . (Square can be used to accept credit cards on your iPhone.) $$\begin{array}{c|c} x & y \\ \hline 0 & 1.0 \\ 2 & 2.0 \\ 7 & 5.5 \\ 12 & 11.0 \end{array}$$ (a) Using ordered pairs, write a function \(T\) that gives the daily transactions in millions of dollars during each month. Interpret the first ordered pair. (b) Repeat part (a) using a diagram. (c) Identify the domain and range of \(T\).

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(5-3 x=x+1\) (b) \(5-3 x \leq x+1\) (c) \(5-3 x \geq x+1\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$a=9, b=10 ; \text { find } c$$

Sketch by hand the graph of the line passing through the given point and having the given slope. Label Through \((-3,0),\) undefined slope}, 2\right)$$,

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(9-(x+1)<0\) (b) \(9-(x+1) \geq 0\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$b=\sqrt{13}, c=\sqrt{29} ; \text { find } a$$

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(6+3(1-x) \geq 0\) (b) \(6+3(1-x)<0\)

Find the length of the unknown side of the right triangle. In each case, a and b represent the lengths of the legs and c represents the length of the hypotenuse. $$b=\sqrt{7}, c=\sqrt{11} ; \text { find } a$$

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(2 x-3>x+2\) (b) \(2 x-3 \leq x+2\)

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(5-3 x \leq-11+x\) (b) \(5-3 x>-11+x\)

Rainfall By noon, 3 inches of rain had fallen during a storm. Rain continued to fall at a rate of \(\frac{1}{4}\) inch per hour. (a) Find a formula for a linear function \(f\) that models the amount of rainfall \(x\) hours past noon. (b) Find the total amount of rainfall by 2: 30 P.M.

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(10 x+5-7 x \geq 8(x+2)+4\) (b) \(10 x+5-7 x<8(x+2)+4\)

U.S. HIV/AIDS Infections In 2010 , there were approximately 1.2 million people in the United States living with HIV/AIDS. At that time the infection rate was \(50,000\) people per year. (a) Find values for \(m\) and \(b\) so that \(y=m x+b\) models the total number of people \(y\) in millions who were living with HIV/AIDS \(x\) years after 2010 . (b) Find \(y\) for the year 2014 . Interpret your result.

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(6 x+2+10 x>-2(2 x+4)+10\) (b) \(6 x+2+10 x \leq-2(2 x+4)+10\)

Distance to Lightning When a bolt of lightning strikes in the distance, there is often a delay between seeing the lightning and hearing the thunder. The function \(f(x)=\frac{x}{5}\) computes the approximate distance in miles between an observer and a bolt of lightning when the delay is \(x\) seconds. (a) Find \(f(15)\) and interpret the result. (b) Graph \(y=f(x) .\) Let the domain of \(f\) be \([0,20]\)

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(5,7), Q(2,11)$$

90\. Air Temperature When the relative humidity is \(100 \%\) air cools \(5.8^{\circ} \mathrm{F}\) for every 1 -mile increase in altitude. If the temperature is \(80^{\circ} \mathrm{F}\) on the ground, then \(f(x)=80-5.8 x\) calculates the air temperature \(x\) miles above the ground. Find \(f(3)\) and interpret the result. (Source: Battan, L., Weather in Your Life, W.H. Freeman.)

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(-2,5), Q(4,-3)$$

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(-11 x-(6 x-4)+5-3 x \leq 1\) (b) \(-11 x-(6 x-4)+5-3 x>1\)

Sales Tax If the sales tax rate is \(7.5 \%,\) write a function \(f\) that calculates the sales tax on a purchase of \(x\) dollars. What is the sales tax on a purchase of \(\$ 86 ?\)

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(-8,-2), Q(-3,-5)$$

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$\frac{1}{3} x-\frac{1}{5} x \leq 2$$

Income and Education Function \(f\) gives the aver- age 2010 individual income (in dollars) by educational attainment for people 25 years old and over. This function is defined by \(f(N)=21,484, f(H)=31,286\) \(f(B)=57,181,\) and \(f(M)=70,181,\) where \(N\) denotes no high school diploma, \(H\) a high school diploma, \(B\) a bachelor's degree, and \(M\) a master's degree. (Source: U.S. Bureau of Labor Statistics.) (a) Write \(f\) as a set of ordered pairs. (b) Give the domain and range of \(f\) (c) Discuss the relationship between education and income.

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(-6,-10), Q(6,5)$$

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$\frac{3 x}{2}+\frac{4 x}{7} \geq-5$$

Tuition and Fees If college tuition costs \(\$ 192\) per credit and fees are fixed at \(\$ 275,\) write a formula for a function \(f\) that calculates the tuition and fees for taking \(x\) credits. What is the total cost of taking 11 credits?

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(9.2,3.4), Q(6.2,7.4)$$

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$0.6 x-2(0.5 x+0.2) \leq 0.4-0.3 x$$

Converting Units of Measure Write a formula for a function \(f\) that converts \(x\) gallons to quarts. How many quarts are there in 19 gallons?

Solve each inequality analytically. Write the solution set in interval notation. Support your answer graphically. $$\frac{2 x+3}{5}-\frac{3 x-1}{2}<\frac{4 x+7}{2}$$

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(8.9,1.6), Q(3.9,13.6)$$