Chapter 7

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. \(\left\\{\begin{array}{l}3 x-2 y=-5 \\ 4 x+y=8\end{array}\right.\)

In Exercises 41-48, graph each horizontal or vertical line. \(y=4\)

Evaluate each function at the given value of the variable. \(h(r)=3 r^{2}+5\) a. \(h(4)\) b. \(h(-1)\)

What is an exponential function?

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 3 . The \(y\)-variable added to the product of 4 and the \(x\)-variable does not exceed 6 .

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. \(\left\\{\begin{array}{l}2 x+5 y=-4 \\ 3 x-y=11\end{array}\right.\)

Graph each horizontal or vertical line. \(y=2\)

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

Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).

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. \(\left\\{\begin{array}{r}x+3 y=2 \\ 3 x+9 y=6\end{array}\right.\)

Graph each horizontal or vertical line. \(y=-2\)

Evaluate each function at the given value of the variable. \(f(x)=2 x^{2}+3 x-1\) a. \(f(3)\) b. \(f(-4)\)

Describe the shape of a scatter plot that suggests modeling the data with a logarithmic function.

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

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. \(\left\\{\begin{array}{l}4 x-2 y=2 \\ 2 x-y=1\end{array}\right.\)

Graph each horizontal or vertical line. \(y=-3\)

Evaluate each function at the given value of the variable. \(f(x)=3 x^{2}+4 x-2\) a. \(f(2)\) b. \(f(-1)\)

Graph each horizontal or vertical line. \(x=2\)

Evaluate each function at the given value of the variable. \(f(x)=\frac{x}{|x|}\) a. \(f(6)\) b. \(f(-6)\)

Describe the shape of a scatter plot that suggests modeling the data with a quadratic function.

Graph each horizontal or vertical line. \(x=4\)

Evaluate each function at the given value of the variable. \(f(x)=\frac{|x|}{x}\) a. \(f(5)\) b. \(f(-5)\)

If \(x\) represents height, in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$ \left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right. $$ Use this information to solve Exercises 45-48. Is a person in this age group who is 6 feet tall weighing 205 pounds within the healthy weight region?

In Exercises 47-48, solve each system for \(x\) and \(y\), expressing either value in terms of a or b, if necessary. Assume that \(a \neq 0\) and \(b \neq 0\). \(\left\\{\begin{array}{l}5 a x+4 y=17 \\ a x+7 y=22\end{array}\right.\)

Graph each horizontal or vertical line. \(x+1=0\)

In Exercises 47-54, evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=x^{2}-1\\\ &\begin{array}{|r|r|} \hline {}{\underline{\phantom{xx}}}{x} & f(x)=x^{2}-1 \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline \end{array} \end{aligned} $$

If \(x\) represents height, in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$ \left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right. $$ Use this information to solve Exercises 45-48. Is a person in this age group who is 5 feet 8 inches tall weighing 135 pounds within the healthy weight region?

Solve each system for \(x\) and \(y\), expressing either value in terms of a or b, if necessary. Assume that \(a \neq 0\) and \(b \neq 0\). \(\left\\{\begin{array}{l}4 a x+b y=3 \\ 6 a x+5 b y=8\end{array}\right.\)

Graph each horizontal or vertical line. \(x+5=0\)

In Exercises 47-54, evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=x^{2}+1\\\ &\begin{array}{|r|l|} \hline {}{\underline{\phantom{xx}}}{x} & f(x)=x^{2}+1 \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline \end{array} \end{aligned} $$

Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Solve each system for \(x\) and \(y\), expressing either value in terms of a or b, if necessary. Assume that \(a \neq 0\) and \(b \neq 0\). For the linear function \(f(x)=m x+b, f(-2)=11\) and \(f(3)=-9\). Find \(m\) and \(b\).

In Exercises 49-52, find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((0, a)\) and \((b, 0)\)

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=x-1\\\ &\begin{array}{|r|l|} \hline {}{\underline{\phantom{xx}}}{\boldsymbol{c}} & \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}-1 \\\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline \end{array} \end{aligned} $$

A patient is not allowed to have more than 330 milligrams of cholesterol per day from a diet of eggs and meat. Each egg provides 165 milligrams of cholesterol. Each ounce of meat provides 110 milligrams. a. Write an inequality that describes the patient's dietary restrictions for \(x\) eggs and \(y\) ounces of meat. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Solve each system for \(x\) and \(y\), expressing either value in terms of a or b, if necessary. Assume that \(a \neq 0\) and \(b \neq 0\). For the linear function \(f(x)=m x+b, f(-3)=23\) and \(f(2)=-7\). Find \(m\) and \(b\).

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=x+1\\\ &\begin{array}{|r|l|} \hline {}{\underline{\phantom{xx}}}{\boldsymbol{x}} & \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}+\mathbf{1} \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline \end{array} \end{aligned} $$

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((a, b)\) and \((a, b+c)\)

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=(x-2)^{2}\\\ &\begin{array}{|l|l|} \hline x & f(x)=(x-2)^{2} \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline 4 & \\ \hline \end{array} \end{aligned} $$

The value of \(a\) in \(y=a x^{2}+b x+c\) and the vertex of the parabola are given. How many \(x\)-intercepts does the parabola have? Explain how you arrived at this number. \(a=1 ;\) vertex at \((2,0)\)

Find the slope of the line passing through each pair of points or state that the slope is undefined. Assume that all variables represent positive real numbers. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. \((a-b, c)\) and \((a, a+c)\)

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=(x+1)^{2}\\\ &\begin{array}{|r|l|} \hline {}{\underline{\phantom{xx}}}{x} & f(x)=(x+1)^{2} \\ \hline-3 & \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline \end{array} \end{aligned} $$

What is a half-plane?

In Exercises 53-54, find the slope and y-intercept of each line whose equation is given. Assume that \(B \neq 0\). \(A x+B y=C\)

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=x^{3}+1\\\ &\begin{array}{|r|c|} \hline {}{\underline{\phantom{xx}}}{x} & f(x)=x^{3}+1 \\ \hline-3 & \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline \end{array} \end{aligned} $$

What does a dashed line mean in the graph of an inequality?

Find the slope and y-intercept of each line whose equation is given. Assume that \(B \neq 0\). \(A x=B y-C\)

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=(x+1)^{3}\\\ &\begin{array}{|r|c|} \hline{}{\underline{\phantom{xx}}}{\boldsymbol{c}} & \boldsymbol{f}(\boldsymbol{x})=(\boldsymbol{x}+\mathbf{1})^{\mathbf{3}} \\ \hline-3 & \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline \end{array} \end{aligned} $$

Explain how to graph \(2 x-3 y<6\).