Chapter 7

Graph each linear inequality. \(y>-4\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x-y=-5 \\ x+5 y=14\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((5,1)\) and \((-2,1)\)

Plot the given point in a rectangular coordinate system. \((1.25,-3.25)\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -4 \\ \hline 1 & -1 \\ \hline 2 & 0 \\ \hline 3 & -1 \\ \hline 4 & -4 \\ \hline \end{array} $$

What is an objective function in a linear programming problem?

Graph each linear inequality. \(y>-2\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{r}2 x+3 y=11 \\ x-4 y=0\end{array}\right.\)

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((-2,3)\) and \((1,3)\)

Plot the given point in a rectangular coordinate system. \((2.25,-4.25)\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -3 \\ \hline 1 & -2 \\ \hline 2 & 0 \\ \hline 3 & 4 \\ \hline 4 & 12 \\ \hline \end{array} $$

What is a constraint in a linear programming problem? How is a constraint represented?

Graph each linear inequality. \(y \geq 0\)

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

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x^{2}-2\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & 4 \\ \hline 1 & 5 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 19 \\ \hline \end{array} $$

In your own words, describe how to solve a linear programming problem.

Graph each linear inequality. \(x>0\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{r}-x+3 y=10 \\ 2 x+8 y=-6\end{array}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x^{2}+2\)

In Exercises 23-24, use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). Based on your graph, describe the shape of a scatter plot that can be modeled by \(f(x)=b^{x}, 0

Describe a situation in your life in which you would like to maximize something, but you are limited by at least two constraints. Can linear programming be used in this situation? Explain your answer.

In Exercises 23-38, graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}3 x+6 y \leq 6 \\ 2 x+y \leq 8\end{array}\right.\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{rr}x+8 y= & 6 \\ 2 x+4 y= & -3\end{array}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x-2\)

Use a table of coordinates to graph each exponential function. Begin by selecting \(-2,-1,0,1\), and 2 for \(x\). Based on your graph, describe the shape of a scatter plot that can be modeled by \(f(x)=b^{x}, 0

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{l}x-y \geq 4 \\ x+y \leq 6\end{array}\right.\)

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{aligned}-4 x+y &=-11 \\ 2 x-3 y &=5 \end{aligned}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x+2\)

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

In Exercises 25-36, solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}x+y=1 \\ x-y=3\end{array}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=2 x+1\)

Use the directions for Exercises 7-8 to graph each logarithmic function. Based on your graph, describe the shape of a scatter plot that can be modeled by \(f(x)=\log _{b} x, 0

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{c}x+y<4 \\ 4 x-2 y<6\end{array}\right.\)

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

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=2 x-4\)

In Exercises 27-28, use the directions for Exercises 9-14 to graph each quadratic function. Use the quadratic formula to find \(x\)-intercepts, rounded to the nearest tenth. \(f(x)=-2 x^{2}+4 x+5\)

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

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

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=-\frac{1}{2} x\)

Use the directions for Exercises 9-14 to graph each quadratic function. Use the quadratic formula to find \(x\)-intercepts, rounded to the nearest tenth. \(f(x)=-3 x^{2}+6 x-2\)

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

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}3 x+2 y=14 \\ 3 x-2 y=10\end{array}\right.\)

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=-\frac{1}{2} x+2\)

In Exercises 29-30, find the vertex for the parabola whose equation is given by writing the equation in the form \(y=a x^{2}+b x+c\).\ \(y=(x-3)^{2}+2\)

Group members should choose a particular field of interest. Research how linear programming is used to solve problems in that field. If possible, investigate the solution of a specific practical problem. Present a report on your findings, including the contributions of George Dantzig, Narendra Karmarkar, and L. G. Khachion to linear programming.

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

Solve each system by the addition method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{r}x+2 y=2 \\ -4 x+3 y=25\end{array}\right.\)

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

In Exercises 29-30, find the vertex for the parabola whose equation is given by writing the equation in the form \(y=a x^{2}+b x+c\).\ \(y=(x-4)^{2}+3\)