Chapter 7

Members of the group should interview a business executive who is in charge of deciding the product mix for a business. How are production policy decisions made? Are other methods used in conjunction with linear programming? What are these methods? What sort of academic background, particularly in mathematics, does this executive have? Present a group report addressing these questions, emphasizing the role of linear programming for the business.

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \geq 4 \\ y \leq 2\end{array}\right.\)

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x-7 y=2 \\ 3 x+y=-20\end{array}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x^{3}-1\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \leq 5 \\ y>-3\end{array}\right.\)

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}4 x+3 y=15 \\ 2 x-5 y=1\end{array}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=|x|+1\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x \leq 3 \\ y>-1\end{array}\right.\)

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}3 x-7 y=13 \\ 6 x+5 y=7\end{array}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=|x|-1\)

The data can be modeled by $$ f(x)=956 x+3176 \text { and } g(x)=3904 e^{0.134 x} \text {, } $$ in which \(f(x)\) and \(g(x)\) represent the average cost of room and board at public four-year colleges in the school year ending \(x\) years after 2010. Use these functions to solve Exercises 33-34. Where necessary, round answers to the nearest whole dollar. a. According to the linear model, what was the average cost of room and board at public four-year colleges for the school year ending in 2017? b. According to the exponential model, what was the average cost of room and board at public four-year colleges for the school year ending in 2017 ? c. Which function is a better model for the data for the school year ending in 2017 ?

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x-y \leq 1 \\ x \geq 2\end{array}\right.\)

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}3 x-4 y=11 \\ 2 x+3 y=-4\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(3 x+y=0\)

In Exercises 33-46, evaluate each function at the given value of the variable. \(f(x)=x-4\) a. \(f(8)\) b. \(f(1)\)

The data can be modeled by $$ f(x)=956 x+3176 \text { and } g(x)=3904 e^{0.134 x} \text {, } $$ in which \(f(x)\) and \(g(x)\) represent the average cost of room and board at public four-year colleges in the school year ending \(x\) years after 2010. Use these functions to solve Exercises 33-34. Where necessary, round answers to the nearest whole dollar. a. According to the linear model, what was the average cost of room and board at public four-year colleges for the school year ending in 2015 ? b. According to the exponential model, what was the average cost of room and board at public four-year colleges for the school year ending in 2015 ? c. Which function is a better model for the data for the school year ending in \(2015 ?\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}4 x-5 y \geq-20 \\ x \geq-3\end{array}\right.\)

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x+3 y=-16 \\ 5 x-10 y=30\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(2 x+y=0\)

In Exercises 33-46, evaluate each function at the given value of the variable. \(f(x)=x-6\) a. \(f(9)\) b. \(f(2)\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}y>2 x-3 \\ y<-x+6\end{array}\right.\)

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x=3 y-4 \\ -6 x+12 y=6\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(3 y=4 x\)

Evaluate each function at the given value of the variable. \(f(x)=3 x-2\) a. \(f(7)\) b. \(f(0)\)

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}y<-2 x+4 \\ y

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}5 x=4 y-8 \\ 3 x+7 y=14\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(4 y=5 x\)

Evaluate each function at the given value of the variable. \(f(x)=4 x-3\) a. \(f(7)\) b. \(f(0)\)

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$ f(x)=62+35 \log (x-4), $$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve Exercises 37-38. a. According to the model, what percentage of her adult height has a girl attained at age 13 ? Use a calculator with a LOG key and round to the nearest tenth of a percent. b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15 , inclusive?

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x+2 y \leq 4 \\ y \geq x-3\end{array}\right.\)

In Exercises 37-44, solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}x=9-2 y \\ x+2 y=13\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(2 x+y=3\)

Evaluate each function at the given value of the variable. \(g(x)=x^{2}+1\) a. \(g(2)\) b. \(g(-2)\)

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$ f(x)=62+35 \log (x-4), $$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve Exercises 37-38. a. According to the model, what percentage of her adult height has a girl attained at age ten? Use a calculator with a LOG key and round to the nearest tenth of a percent. b. Why was a logarithmic function used to model the percentage of adult height attained by a girl from ages 5 to 15 , inclusive?

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x+y \leq 4 \\ y \geq 2 x-4\end{array}\right.\)

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}6 x+2 y=7 \\ y=2-3 x\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(3 x+y=4\)

Evaluate each function at the given value of the variable. \(g(x)=x^{2}+4\) a. \(g(3)\) b. \(g(-3)\)

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { x, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y} \text {, Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 1 & 7.6 \\ \hline 3 & 6 \\ \hline 4 & 2.8 \\ \hline \end{array} $$ QuadReg $$ \begin{aligned} &y=a x^{2}+b x+c \\ &a=-.8 \\ &b=2.4 \\ &c=6 \end{aligned} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs feet from where it was thrown and the maximum height is feet.

In Exercises 39-40, write each sentence as an inequality in two variables. Then graph the inequality. The \(y\)-variable is at least 4 more than the product of \(-2\) and the \(x\)-variable.

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}y=3 x-5 \\ 21 x-35=7 y\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(7 x+2 y=14\)

Evaluate each function at the given value of the variable. \(g(x)=-x^{2}+2\) a. \(g(4)\) b. \(g(-3)\)

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { x, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y} \text {, Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 0.5 & 7.4 \\ \hline 1.5 & 9 \\ \hline 4 & 6 \\ \hline \end{array} $$ QuadReg \(y=a x^{2}+b x+c\) \(a=-.8\) \(\mathrm{b}=3.2\) \(c=6\)

Write each sentence as an inequality in two variables. Then graph the inequality. The \(y\)-variable is at least 2 more than the product of \(-3\) and the \(x\)-variable.

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}9 x-3 y=12 \\ y=3 x-4\end{array}\right.\)

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(5 x+3 y=15\)

Evaluate each function at the given value of the variable. \(g(x)=-x^{2}+1\) a. \(g(5)\) b. \(g(-4)\)

What is a scatter plot?

In Exercises 41-42, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the \(x\)-variable and the \(y\)-variable is at most 4 . The \(y\)-variable added to the product of 3 and the \(x\)-variable does not exceed \(6 .\)