Chapter 5

will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ 2 x-3 y=6 $$

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?

When a crew rows with the current, it travels 16 miles in 2 hours. Against the current, the crew rows 8 miles in 2 hours. Let \(x=\) the crew's rowing rate in still water and let \(y=\) the rate of the current. The following chart summarizes this information: Find the rate of rowing in still water and the rate of the current. (TABLE CAN'T COPY)

will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ f(x)=-\frac{2}{3} x $$

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?

When an airplane flies with the wind, it travels 800 miles in 4 hours. Against the wind, it takes 5 hours to cover the same distance. Find the plane's rate in still air and the rate of the wind.

will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ f(x)=-2 $$

On your next vacation, you will divide lodging between large resorts and small inns. Let \(x\) represent the number of nights spent in large resorts. Let \(y\) represent the number of nights spent in small inns. a. Write a system of inequalities that models the following conditions:You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average \(\$ 200\) per night and small inns average \(\$ 100\) per night. Your budget permits no more than \(\$ 700\) for lodging. b. Graph the solution set of the system of inequalities in part (a). c. Based on your graph in part (b), what is the greatest number of nights you could spend at a large resort and still stay within your budget?

A person with no more than \(\$ 15,000\) to invest plans to place the money in two investments. One investment is high risk, high yield; the other is low risk, low yield. At least \(\$ 2000\) is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least three times the amount invested at high risk. Find and graph a system of inequalities that describes all possibilities for placing the money in the high- and low-risk investments.

What is a system of linear equations? Provide an example with your description.

What is a linear inequality in two variables? Provide an example with your description.

Explain how to solve a system of equations using the substitution method. Use \(y=3-3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

How do you determine if an ordered pair is a solution of an inequality in two variables, \(x\) and \(y ?\)

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

What is a half-plane?

When is it easier to use the addition method rather than the substitution method to solve a system of equations?

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

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

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

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations?

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\) Discuss similarities and differences between the graphs.

What is a system of linear inequalities?

What is a solution of a system of linear inequalities?

Explain how to graph the solution set of a system of inequalities.

What does it mean if a system of linear inequalities has nosolution?

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$y \leq 4 x+4$$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$ y \geq \frac{2}{3} x-2 $$

Write a system of equations having \(\\{(-2,7)\\}\) as a solution set. (More than one system is possible.)

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$ y \geq x^{2}-4 $$

Solve the system for \(x\) and \(y\) in terms of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}\) and \(c_{2}\) $$ \left\\{\begin{array}{l} a_{1} x+b_{1} y=c_{1} \\ a_{2} x+b_{2} y=c_{2} \end{array}\right. $$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$ y \geq \frac{1}{2} x^{2}-2 $$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$2 x+y \leq 6$$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. $$3 x-2 y \geq 6$$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities. Does your graphing utility have any limitations in terms of graphing inequalities? If so, what are they?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing a linear inequality, I should always use \((0,0)\) as a test point because it's easy to perform the calculations when 0 is substituted for each variable.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12,\) it's not necessary for me to graph the linear equation \(3 x-4 y=12\) because the inequality contains a \(<\) symbol, in which equality is not included.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The reason that systems of linear inequalities are appropriate for modeling healthy weight is because guidelines give healthy weight ranges, rather than specific weights, for various heights.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed the solution set of \(y \geq x+2\) and \(x \geq 1\) without using test points.

Write a system of inequalities whose solution set includes every point in the rectangular coordinate system.

Sketch the graph of the solution set for the following system of inequalities: $$\left\\{\begin{array}{l}y \geq n x+b(n<0, b>0) \\\y \leq m x+b(m>0, b>0)\end{array}\right.$$

Will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{aligned}x+y & \geq 6 \\\x & \leq 8 \\\y & \geq 5\end{aligned}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(3 x+2 y\) at each of the points obtained in part (b)

Will help you prepare for the material covered in the next section. a. Graph the solution set of the system:$$\left\\{\begin{array}{c}x \geq 0 \\\y \geq 0 \\\3 x-2 x \leq 6 \\\y \leq-x+7\end{array}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(2 x+5 y\) at each of the points obtained in part (b)

Will help you prepare for the material covered in the next section. Bottled water and medical supplies are to be shipped to survivors of an earthquake by plane. The bottled water weighs 20 pounds per container and medical kits weigh 10 pounds per kit. Each plane can carry no more than \(80,000\) pounds. If \(x\) represents the number of bottles of water to be shipped per plane and \(y\) represents the number of medical kits per plane, write an inequality that models each plane's \(80,000\) -pound weight restriction.