Chapter 2

Exercises 75 and 76 : Graph \(f\). $$ f(x)=\left\\{\begin{array}{ll} -\frac{1}{2} x+1 & \text { if }-4 \leq x \leq-2 \\ 1-2 x & \text { if }-2

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ -\frac{3}{4}<\frac{2-5 x}{3} \leq \frac{3}{4} $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 7-(3-2 x)=1 $$

Exercises 75 and 76 : Graph \(f\). $$ f(x)=\left\\{\begin{array}{ll} \frac{3}{2}-\frac{1}{2} x & \text { if }-3 \leq x<-1 \\ -2 x & \text { if }-1 \leq x \leq 2 \\ \frac{1}{2} x-5 & \text { if } 2

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ \frac{3 x-1}{5}<15 $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ \sqrt{3}(2-\pi x)+x=0 $$

Use \(f(x)\) to complete the following: $$ f(x)=\left\\{\begin{array}{ll} 3 x-1 & \text { if }-5 \leq x<1 \\ 4 & \text { if } 1 \leq x \leq 3 \\ 6-x & \text { if } \quad 3

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ (\sqrt{11}-\pi) x-5.5 \leq 0 $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 3(\pi-x)+\sqrt{2}=0 $$

Use \(g(x)\) to complete the following. $$ g(x)=\left\\{\begin{array}{ll} -2 x-6 & \text { if }-8 \leq x \leq-2 \\ x & \text { if }-2

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ 1.5(x-0.7)+1.5 x<1 $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ x-3=2 x+1 $$

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ 2 x-8>5 $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 3(x-1)=2 x-1 $$

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ 5<4 x-2.5 $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 6 x-8=-7 x+18 $$

In 2003 the average amual cost of attending a private college or university, including tuition, fecs, room, and board, was \(\$ 25,000 .\) This cost is projected to rise to \(\$ 37,000\) in 2010 , as illustrated in the figure. (Source: Cerulli Associates.) (GRAPH CANNOT COPY) (A) Find a point-slope form of the line passing through the points \((2003,25000)\) and \((2010,37000)\). Interpret the slope. (B) Use the equation to estimate the cost of altending a private college in \(2007 .\) Did your estimate involve interpolation or extrapolation? (C) Find the slope-intercept form of this line.

Exercises \(79-82:\) Complete the following. (a) Use dot mode to graph the function \(f\) in the standard viewing rectangle (b) Evaluate \(f(-3.1)\) and \(f(1.7)\) $$ f(x)=2[x]+1 $$

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ \pi x-5.12 \leq \sqrt{2} x-5.7(x-1.1) $$

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 5-8 x=3(x-7)+37 $$

Exercises \(79-82:\) Complete the following. (a) Use dot mode to graph the function \(f\) in the standard viewing rectangle (b) Evaluate \(f(-3.1)\) and \(f(1.7)\) $$ f(x)=[-x] $$

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ 5.1 x-\pi \geq \sqrt{3}-1.7 x $$

Exercises \(83-90:\) Solve the equation for the specified variable. \(A=L W\) for \(W\)

In 2002 sales of premium online music totaled \(\$ 1.6\) billion. In 2005 this revenue reached \(\$ 3.6\) billion. (A) Find a point-slope form of the line passing through \((2002,1.6)\) and \((2005,3.6) .\) Interpret the slope. (B) Use the equation to estimate projected sales in2008. Did you use interpolation or extrapolation? (C) Find the slope-intercept form of this line.

Lumber costs The lumber used to frame walls of houses is frequently sold in multiples of 2 feet. If the length of a board is not exactly a multiple of 2 feet, there is often no charge for the additional length. For example, if a board measures at least 8 feet but less than 10 feet, then the consumer is charged for only 8 feet. (a) Suppose that the cost of lumber is \(\$ 0.80\) for every 2 feet. Find a formula for a function \(f\) that computes the cost of a board \(x\) feet long for \(6 \leq x \leq 18\) (b) Graph \(f\) (c) Determine the costs of boards with lengths of 8.5 feet and 15.2 feet.

Solve the equation for the specified variable. $$ E=I R+2 \text { for } R $$

cost of Carpet Each foot of carpet purchased from a 12-foot-wide roll costs \(\$ 36 .\) If a fraction of a foot is purchased, a customer does not pay for the extra amount. For example, if a customer wants 14 feet of carpet and the salesperson cuts off 14 feet 4 inches, the customer does not pay for the extra 4 inches. (a) How much does 9 feet 8 inches of carpet from this roll cost? (b) Using the greatest integer function, write a formula for the price \(P\) of \(x\) feet of carpet.

Solve the equation for the specified variable. $$ P=2 L+2 W \text { for } L $$

Exercises 85 and 86: Find the line of least-squares fit for the given data points. What is the correlation coefficient? Plot the data and graph the line. $$ (-2,2),(1,0),(3,-2) $$

Solve the equation for the specified variable. $$ V=2 \pi r h+\pi r^{2} \text { for } h $$

Exercises 85 and 86: Find the line of least-squares fit for the given data points. What is the correlation coefficient? Plot the data and graph the line. $$ (-1,-1),(1,4),(2,6) $$

Solve the equation for the specified variable. $$ 3 x+2 y=8 \text { for } y $$

Exercises \(87-90:\) Complete the following. (a) Conjecture whether the correlation coefficient \(r\) for the data will be positive, negative, or zero. (b) Use a calculator to find the equation of the least squares regression line and the value of \(\boldsymbol{r}\). (c) Use the regression line to predict y when \(x=2.4\) $$ \begin{array}{cccccc} x & -1 & 0 & 1 & 2 & 3 \\ \hline y & -5.7 & -2.6 & 1.1 & 3.9 & 7.3 \end{array} $$

Prices of Homes The median prices of a single-family home in the United States from 1990 to 2005 can be approximated by the formula \(P(x)=8667 x+90,000\) where \(x=0\) corresponds to 1990 and \(x=15\) to 2005 (Source: National Association of Realtors.) (a) Interpret the slope of the graph of \(P\). (b) Estimate the years when the median price range was from \(\$ 142,000\) to \(\$ 194,000\)

Solve the equation for the specified variable. $$ 5 x-4 y=20 \text { for } y $$

In 1988 the number of farm pollution incidents reported in England and Wales was \(4000 .\) This number had increased at a rate of 280 per year since 1979\. (Source: C. Mason, Biology of Freshwater Pollution.) (a) Find an equation \(y=m\left(x-x_{1}\right)+y_{1}\) that models these data, where \(y\) represents the number of pollution incidents during the year \(x\) (b) Estimate the number of incidents in 1975 .

Exercises \(87-90:\) Complete the following. (a) Conjecture whether the correlation coefficient \(r\) for the data will be positive, negative, or zero. (b) Use a calculator to find the equation of the least squares regression line and the value of \(\boldsymbol{r}\). (c) Use the regression line to predict y when \(x=2.4\) $$ \begin{array}{cccccc} x & -4 & -2 & 0 & 2 & 4 \\ \hline y & 1.2 & 2.8 & 5.3 & 6.7 & 9.1 \end{array} $$

Population Density The population density \(D\) of the United States in people per square mile during year \(x\) from 1900 to 2000 can be approximated by the formula \(D(x)=0.58 x-1080 . \quad\) (Source: Bureau of the Census.) (a) Interpret the slope of the graph of \(D\). (b) Estimate when the density varied between 50 and 75 people per square mile.

Solve the equation for the specified variable. $$ y=3(x-2)+x \text { for } x $$

The cost of driving a car includes both fixed costs and mileage costs. Assume that insurance and car payments cost \(\$ 350\) per month and gasoline, oil, and routine maintenance cost \(\$ 0.29\) per mile. (a) Find a linear function \(f\) that gives the annual cost of driving this car \(x\) miles. (b) What does the \(y\) -intercept on the graph of \(f\) represent?

Exercises \(87-90:\) Complete the following. (a) Conjecture whether the correlation coefficient \(r\) for the data will be positive, negative, or zero. (b) Use a calculator to find the equation of the least squares regression line and the value of \(\boldsymbol{r}\). (c) Use the regression line to predict y when \(x=2.4\) $$ \begin{array}{cccccc} x & 1 & 3 & 5 & 7 & 10 \\ \hline y & 5.8 & -2.4 & -10.7 & -17.8 & -29.3 \end{array} $$

Broadband Internet Connections The number of households using broadband Internet connections, such as cable and DSL, increased from 6 million in 2000 to 30 million in 2004 . (Source: eMarketer.) (a) Find a linear function given by $$ B(x)=m\left(x-x_{1}\right)+y_{1} $$ that models these data, where \(x\) is the year. (b) Use \(B(x)\) to estimate the years when the number of households using broadband Intemet connections was 24 million or more. Assume that the domain of \(B\) is 2000 to 2006

Solve the equation for the specified variable. $$ y=4-(8-2 x) \text { for } x $$

The average hourly wage (adjusted to 1982 dollars) was \(\$ 8.46\) in 1970 and \(\$ 8.18\) in 2005 (Source: Department of Commerce.) (A) Find an equation of a line that passes through the points \((1970,8.46)\) and \((2005,8.18)\) (B) Interpret the slope. (C) Approximate the hourly wage in 2000 . Compare the estimate to the actual value of \(\$ 8.04\)

Exercises \(87-90:\) Complete the following. (a) Conjecture whether the correlation coefficient \(r\) for the data will be positive, negative, or zero. (b) Use a calculator to find the equation of the least squares regression line and the value of \(\boldsymbol{r}\). (c) Use the regression line to predict y when \(x=2.4\) $$ \begin{array}{cccccc} x & -4 & -3 & -1 & 3 & 5 \\ \hline y & 37.2 & 33.7 & 27.5 & 16.4 & 9.8 \end{array} $$

Online Betting Consumer gambling losses from online betting were \(\$ 4\) billion in 2002 and \(\$ 10\) billion in \(2005 . \quad\) (Source: Christiansen Capital Advisors.) (a) Find a linear function given by $$ B(x)=m\left(x-x_{1}\right)+y_{1} $$ that models these data, where \(x\) is the year. (b) Use \(B(x)\) to estimate the years when consumer losses from online betting were more than \(\$ 6\) billion. Assume that the domain of \(B\) is 2002 to 2007 .

Income The per capita (per person) income from 1980 to 2006 can be modeled by $$ f(x)=1000(x-1980)+10,000 $$ where \(x\) is the year. Determine the year when the per capita income was \(\$ 19,000 .\) (Source: Bureau of the Census.)

Consumer Spending In 2005 consumers used credit and debit cards to pay for \(40 \%\) of all purchases. This percentage is projected to be \(55 \%\) in 2011 . (Source: Bloomburg.) (a) Find a linear function \(P\) that models the data. (b) Estimate when this percentage was between \(45 \%\) and \(50 \%\)

VISA Cards Annual transactions on VISA cards increased from \(\$ 400\) billion in 2002 to \(\$ 635\) billion in \(2007 . \quad\) (Source: CardWeb.) (a) Find a linear function \(V\) that models the data. (b) Estimate when this number was between \(\$ 450\) billion and \(\$ 540\) billion.

Vinyl and CD Sales During the 1980 s, sales of compact discs surpassed vinyl record sales. From 1985 to 1990 , sales of compact discs in millions can be modeled by the formula \(f(x)=51.6(x-1985)+9.1,\) whereas sales of vinyl LP records in millions can be modeled by \(g(x)=-31.9(x-1985)+167.7 .\) Approximate the year \(x\) when sales of LP records and compact discs were equal by using the intersection-of-graphs method. (Source: Recording Industry Association of America.)