1/9/2018 thinking mathematically, sixth edition · 2018. 8. 31. · part of the solution set. thus,...

1/9/2018 Thinking Mathematically, Sixth Edition

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7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables

7.4 Linear Inequalities in Two Variables

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Graph a linear inequality in two variables.

2 Use mathematical models involving linear inequalities.

3 Graph a system of linear inequalities.

WE OPENED THE CHAPTER NOTING THAT THE modern emphasis on thinness as the ideal body shape has been suggested as a major cause of eating disorders. Inthis section (Example 4), as well as in the Exercise Set (Exercises 45–48), we use systems of linear inequalities in two variables that will enable you to establish ahealthy weight range for your height and age.

Linear Inequalities in Two Variables and Their SolutionsWe have seen that equations in the form where A and B are not both zero, are straight lines when graphed. If we change the or we obtain a linear inequality in two variables. Some examples of linear inequalities in two variables are and

A solution of an inequality in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and they-coordinate is substituted for y in the inequality, we obtain a true statement. For example, is a solution of the inequality When 3 is substituted for xand 2 is substituted for y, we obtain the true statement or Because there are infinitely many pairs of numbers that have a sum greater than 1, theinequality has infinitely many solutions. Each ordered-pair solution is said to satisfy the inequality. Thus, satisfies the inequality

The Graph of a Linear Inequality in Two Variables

1 Graph a linear inequality in two variables.

We know that the graph of an equation in two variables is the set of all points whose coordinates satisfy the equation. Similarly, the graph of an inequality in twovariables is the set of all points whose coordinates satisfy the inequality.

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Ax + By = C, symbol = to >, <, ≥ ,≤ , x + y > 2, 3x − 5y ≤ 15, 2x − y < 4.

(3, 2) x + y > 1.3 + 2 > 1, 5 > 1.

x + y > 1 (3, 2) x + y > 1.

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Let's use Figure 7.34 to get an idea of what the graph of a linear inequality in two variables looks like. Part of the figure shows the graph of the linear equation The line divides the points in the rectangular coordinate system into three sets. First, there is the set of points along the line satisfying Next,

there is the set of points in the green region above the line. Points in the green region satisfy the linear inequality Finally, there is the set of points in thepurple region below the line. Points in the purple region satisfy the linear inequality

FIGURE 7.34

A half-plane is the set of all the points on one side of a line. In Figure 7.34, the green region is a half-plane. The purple region is also a half-plane. A half-plane is thegraph of a linear inequality that involves or The graph of an inequality that involves or is a half-plane and a line. A solid line is used to show that a line ispart of a graph. A dashed line is used to show that a line is not part of a graph.

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x + y = 2. x + y = 2.x + y > 2.

x + y < 2.

> < . ≥ ≤



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7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables

Graphing a Linear Inequality in Two Variables1. Replace the inequality symbol with an equal sign and graph the corresponding linear equation. Draw a solid line if the original inequality contains a or symbol. Draw a dashed line if the original inequality contains a or symbol.

2. Choose a test point from one of the half-planes. (Do not choose a point on the line.) Substitute the coordinates of the test point into the inequality.

3. If a true statement results, shade the half-plane containing this test point. If a false statement results, shade the half-plane not containing this test point.

Example 1 Graphing a Linear Inequality in Two VariablesGraph:

SOLUTION

Step 1 Replace the inequality symbol graph the linear equation.

We need to graph We can use intercepts to graph this line.

We set to find the x-intercept. We set to find the y-intercept.

The x-intercept is 5, so the line passes through The y-intercept is so the line passes through Using the intercepts, the line is shown in Figure7.35 as a solid line. The line is solid because the inequality contains a symbol, in which equality is included.

FIGURE 7.35 Preparing to graph

Step 2 Choose a test point from one of the half-planes and not from the line. Substitute its coordinates into the inequality. The line dividesthe plane into three parts—the line itself and two half-planes. The points in one half-plane satisfy The points in the other half-plane satisfy

We need to find which half-plane belongs to the solution of To do so, we test a point from either half-plane. The origin, is theeasiest point to test.

Step 3 If a false statement results, shade the half-plane not containing the test point. Because 0 is not greater than or equal to 15, the test point, is notpart of the solution set. Thus, the half-plane below the solid line is part of the solution set. The solution set is the line and the half-plane that does notcontain the point indicated by shading this half-plane. The graph is shown using green shading and a blue line in Figure 7.36.



≤ ≥< >

3x − 5y ≥ 15.

by = and

3x − 5y = 15.

y = 0 x = 0

3x − 5y = 15

3x − 5 ⋅ 0 = 15

3x = 15

x = 5

3x − 5y = 15

3.0 − 5y = 15

−5y = 15

y = −3(5, 0) . −3, (0,−3) .

3x − 5y ≥ 15 ≥

3x − 5y ≥ 15

3x − 5y = 153x − 5y > 15.

3x − 5y < 15. 3x − 5y ≥ 15. (0, 0) ,

3x − 5y

3 ⋅ 0 − 5 ⋅ 0

0 − 0

0

≥

≥?

≥?

≥

15

15

15

15

This is the given inequality.

Test (0, 0) by substituting 0 for x and 0 for y.

Multiply.

This statement is false.

(0, 0) ,3x − 5y = 15

(0, 0) ,











48

FIGURE 7.36 The graph of

Check Point 1Graph:


3x − 5y ≥ 15

2x − 4y ≥ 8.








7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables > Graphing Linear Inequalitieswithout Using Test Points

When graphing a linear inequality, test a point that lies in one of the half-planes and not on the line separating the half-planes. The test point is convenientbecause it is easy to calculate when 0 is substituted for each variable. However, if lies on the dividing line and not in a half-plane, a different test point must beselected.

Example 2 Graphing a Linear Inequality in Two VariablesGraph:

SOLUTION

Step 1 Replace the inequality symbol graph the linear equation. Because we are interested in graphing we begin by graphing

We can use the slope and the y-intercept to graph this linear function.

The y-intercept is 0, so the line passes through (0, 0). Using the y-intercept and the slope, the line is shown in Figure 7.37 as a dashed line. The line is dashedbecause the inequality contains a symbol, in which equality is not included.

dFIGURE 7.37 The graph of

Step 2 Choose a test point from one of the half-planes and not from the line. Substitute its coordinates into the inequality. We cannot use as a testpoint because it lies on the line and not in a half-plane. Let's use which lies in the half-plane above the line.

Step 3 If a true statement results, shade the half-plane containing the test point. Because 1 is greater than the test point, is part of the solution

set. All the points on the same side of the line as the point are members of the solution set. The solution set is the half-plane that contains thepoint indicated by shading this half-plane. The graph is shown using green shading and a dashed blue line in Figure 7.37.

Check Point 2Graph:

Graphing Linear Inequalities without Using Test PointsYou can graph inequalities in the form or without using test points. The inequality symbol indicates which half-plane to shade.

• If shade the half-plane above the line

• If shade the half-plane below the line

Observe how this is illustrated in Figure 7.37. The graph of is the half-plane above the line



(0, 0)(0, 0)

y > − x.23

by = and y > − x,23

y = − x.23

y = − x + 0 ◄ 23

y-intercept = 0

▲

Slope = =−2

3riserun

y > − x23

>

y > − x23

(0, 0)(1, 1) ,

y > − x23

1 − ⋅ 1>? 2

3

1 > − 23

This is the given inequality.

Test (1, 1)by substituting 1 for x and 1 fory.

This statement is true.

− ,23

(1, 1) ,

y = − x23

(1, 1)(1, 1) ,

y > − x.3

4

y > mx + b y < mx + b

y > mx + b, y = mx + b.

y < mx + b, y = mx + b.

y > − x23

y = − x.23












Page 449









7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables > Modeling with Systems of LinearInequalities

It is also not necessary to use test points when graphing inequalities involving half-planes on one side of a vertical or a horizontal line.

For the Vertical Line For the Horizontal Line • If shade the half-plane to the right of • If shade the half-plane above • If shade the half-plane to the left of • If shade the half-plane below

Great Question!When is it important to use test points to graph linear inequalities?

Continue using test points to graph inequalities in the form or The graph of can lie above or below the line givenby depending on the values of A and B. The same comment applies to the graph of

Example 3 Graphing Inequalities without Using Test PointsGraph each inequality in a rectangular coordinate system:

a.

b.

SOLUTION

a.

b.



x = a: y = b:

x > a, x = a. y > b, y = b.

x < a, x = a. y < b, y = b.

Ax + By > C Ax + By < C. Ax + By > CAx + By = C, Ax + By < C.

y ≤ −3

x > 2.

y ≤ −3

x > 2












Page 450

Check Point 3Graph each inequality in a rectangular coordinate system:

a.

b.

Modeling with Systems of Linear Inequalities

2 Use mathematical models involving linear inequalities.

Just as two or more linear equations make up a system of linear equations, two or more linear inequalities make up a system of linear inequalities. A solution of asystem of linear inequalities in two variables is an ordered pair that satisfies each inequality in the system.


y > 1

x ≤ −2.








Page 451

7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables > Modeling with Systems of LinearInequalities

Example 4 Does Your Weight Fit You?The latest guidelines, which apply to both men and women, give healthy weight ranges, rather than specific weights, for your height. Figure 7.38 shows the healthyweight region for various heights for people between the ages of 19 and 34, inclusive.

dFIGURE 7.38 Source: U.S. Department of Health and Human Services

If x represents height, in inches, and y represents weight, in pounds, the healthy weight region in Figure 7.38 can be modeled by the following system of linearinequalities:

Show that point A in Figure 7.38 is a solution of the system of inequalities that describes healthy weight.

SOLUTION

Point A has coordinates This means that if a person is 70 inches tall, or 5 feet 10 inches, and weighs 170 pounds, then that person's weight is within thehealthy weight region. We can show that satisfies the system of inequalities by substituting 70 for x and 170 for y in each inequality in the system.

The coordinates make each inequality true. Thus, satisfies the system for the healthy weight region and is a solution of the system.

Check Point 4Show that point B in Figure 7.38 is a solution of the system of inequalities that describes healthy weight.




{ 4.9x − y ≥ 165

3.7x − y ≤ 125.

(70, 170) .(70, 170)

4.9x − y ≥ 165

4.9 (70) − 170 ≥ 165

343 − 170 ≥ 165

173 ≥ 165,

true

3.7x − y ≤ 125

3.7 (70) − 170 ≤ 125

259 − 170 ≤ 125

89 ≤ 125,

true

(70, 170) (70, 170)

















Page 452

7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables > Graphing Systems of LinearInequalities

Graphing Systems of Linear Inequalities

3 Graph a system of linear inequalities.

The solution set of a system of linear inequalities in two variables is the set of all ordered pairs that satisfy each inequality in the system. Thus, to graph a system ofinequalities in two variables, begin by graphing each individual inequality in the same rectangular coordinate system. Then find the region, if there is one, that is commonto every graph in the system. This region of intersection gives a picture of the system's solution set.

Example 5 Graphing a System of Linear InequalitiesGraph the solution set of the system:

SOLUTION

Replacing each inequality symbol in and with an equal sign indicates that we need to graph and We can useintercepts to graph these lines.

The line passes through The line passes through

The line passes through The line passes through Now we are ready to graph the solution set of the system of linear inequalities.

d d




{ x − y

2x + 3y

< 1

≥ 12.

x − y < 1 2x + 3y ≥ 12 x − y = 1 2x + 3y = 12.

x − y = 1 2x + 3y = 12

x-intercept : x − 0 = 1

x = 1◄ ►Set y = 0 in each equation.

x-intercept: 2x + 3 ⋅ 0 = 12

2x = 12

x = 6

(1, 0). (6, 0).

y-intercept : 0 − y = 1

−y = 1

y = −1

◄ ►Set x = 0 in each equation.

y-intercept: 2 ⋅ 0 + 3y = 12

3y = 12

y = 4

(0, −1). (0, 4).

















7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables > Concept and Vocabulary Check

Check Point 5Graph the solution set of the system:

Example 6 Graphing a System of Linear InequalitiesGraph the solution set of the system:

SOLUTION

d

Check Point 6Graph the solution set of the system:

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. The ordered pair (3, 2) is a/an ___________________ of the inequality because when 3 is substituted for ___________________ and 2 is substitutedfor ___________________, the true statement ___________________ is obtained.

2. The set of all points that satisfy a linear inequality in two variables is called the ___________________ of the inequality.

3. The set of all points on one side of a line is called a/an ___________________.

4. True or False: The graph of includes the line ___________________

5. True or False: The graph of the linear equation is used to graph the linear inequality ___________________

6. True or False: When graphing to determine which side of the line to shade, choose a test point on ___________________

7. The solution set of the system

is the set of ordered pairs that satisfy ___________________ and _____________________.



{ x + 2y > 4

2x − 3y ≤ −6.

{x ≤ 4

y > −2.

{x < 3

y ≥ −1.

x + y > 1

2x − 3y > 6 2x − 3y = 6.

2x − 3y = 6 2x − 3y > 6.

4x − 2y ≥ 8, 4x − 2y = 8.

{ x − y < 1

2x + 3y ≥ 12












Page 453

8. True or False: The graph of the solution set of the system

includes the intersection point of and ___________________


{ x − 3y < 6

2x + 3y ≥ −6

x − 3y = 6 2x + 3y = −6.








7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables > Exercise Set 7.4

Exercise Set 7.4Practice ExercisesIn Exercises 1–22, graph each linear inequality.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

In Exercises 23–38, graph the solution set of each system of inequalities.

23.

24.

25.

26.

27.

28.

29.

30.



x + y ≥ 2

x − y ≤ 1

3x − y ≥ 6

3x + y ≤ 3

2x + 3y > 12

2x − 5y < 10

5x + 3y ≤ −15

3x + 4y ≤ −12

2y − 3x > 6

2y − x > 4

y > x1

3

y > x1

4

y ≤ 3x + 2

y ≤ 2x − 1

y < − x1

4

y < − x1

3

x ≤ 2

x ≤ −4

y > −4

y > −2

y ≥ 0

x ≥ 0

{3x + 6y ≤ 6

2x + y ≤ 8

{x − y ≥ 4

x + y ≤ 6

{2x + y < 3

x − y > 2

{ x + y < 4

4x − 2y < 6

{2x + y < 4

x − y > 4

{2x − y < 3

x + y < 6

{ x ≥ 2

y ≤ 3

{ x ≥ 4

y ≤ 2












31.

32.

33.

34.

35.

36.

37.

38.

Practice PlusIn Exercises 39–40, write each sentence as an inequality in two variables. Then graph the inequality.

39. The y-variable is at least 4 more than the product of and the x-variable.

40. The y-variable is at least 2 more than the product of and the x-variable.

In Exercises 41–42, write the given sentences as a system of inequalities in two variables. Then graph the system.

41. 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.

42. 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.

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, youwill be graphing the union of the solution sets of two inequalities.

43. Graph the union of and

44. Graph the union of and

Application ExercisesThe figure shows the healthy weight region for various heights for people ages 35 and older.

dSource: U.S. Department of Health and Human Services

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:

Use this information to solve Exercises 45–48.

45. Show that point A is a solution of the system of inequalities that describes healthy weight for this age group.

46. Show that point B is a solution of the system of inequalities that describes healthy weight for this age group.

47. Is a person in this age group who is 6 feet tall weighing 205 pounds within the healthy weight region?

{x ≤ 5

y > −3

{x ≤ 3

y > −1

{x − y ≤ 1

x ≥ 2

{4x − 5y ≥ −20

x ≥ −3

{y > 2x − 3

y < −x + 6

{y < −2x + 4

y < x − 4

{x + 2y ≤ 4

y ≥ x − 3

{x + y ≤ 4

y ≥ 2x − 4

−2

−3

y > x − 23

2y < 4.

x − y ≥ −1 5x − 2y ≤ 10.

{ 5.3x − y ≥ 180

4.1x − y ≤ 140.



Page 454

48. Is a person in this age group who is 5 feet 8 inches tall weighing 135 pounds within the healthy weight region?

49. 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?









7 Algebra: Graphs, Functions and Linear Systems > 7.4 Linear Inequalities in Two Variables > Exercise Set 7.4

50. 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?

The graph of an inequality in two variables is a region in the rectangular coordinate system. Regions in coordinate systems have numerous applications. For example,the regions in the following two graphs indicate whether a person is obese, overweight, borderline overweight, normal weight, or underweight.

Expand this image

dSource: Centers for Disease Control and Prevention

In these graphs, each horizontal axis shows a person's age. Each vertical axis shows that person's body-mass index (BMI), computed using the following formula:

The variable W represents weight, in pounds. The variable H represents height, in inches. Use this information and the graphs shown above to solve Exercises 51–52.

51. A man is 20 years old, 72 inches (6 feet) tall, and weighs 200 pounds.

a. Compute the man's BMI. Round to the nearest tenth.

b. Use the man's age and his BMI to locate this information as a point in the coordinate system for males. Is this person obese, overweight, borderline overweight,normal weight, or underweight?

52. A woman is 25 years old, 66 inches (5 feet, 6 inches) tall, and weighs 105 pounds.

a. Compute the woman's BMI. Round to the nearest tenth.

b. Use the woman's age and her BMI to locate this information as a point in the coordinate system for females. Is this person obese, overweight, borderlineoverweight, normal weight, or underweight?

Writing in Mathematics53. What is a half-plane?

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

55. Explain how to graph

56. Compare the graphs of and Discuss similarities and differences between the graphs.

57. Describe how to solve a system of linear inequalities.

Critical Thinking ExercisesMake Sense? In Exercises 58–61, determine whether each statement makes sense or does not make sense, and explain your reasoning.

58. 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.

59. When graphing it's not necessary for me to graph the linear equation because the inequality contains a symbol, in whichequality is not included.

60. Systems of linear inequalities are appropriate for modeling healthy weight because guidelines give healthy weight ranges, rather than specific weights, for variousheights.



BMI = .703W

H 2

2x − 3y < 6.

3x − 2y > 6 3x − 2y ≤ 6.

3x − 4y < 12, 3x − 4y = 12 <




javascript:fullImageLayer("NIMAS0321867327-img-951")









Page 455

61. I graphed the solution set of and without using test points.

In Exercises 62–63, write a system of inequalities for each graph.

62.

63.


y ≥ x + 2 x ≥ 1








7 Algebra: Graphs, Functions and Linear Systems > 7.5 Linear Programming

Without graphing, in Exercises 64–67, determine if each system has no solution or infinitely many solutions.

64.

65.

66.

67.

7.5 Linear Programming

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Write an objective function describing a quantity that must be maximized or minimized.

2 Use inequalities to describe limitations in a situation.

3 Use linear programming to solve problems.

d

THE BERLIN AIRLIFT (1948–1949) was an operation by the United States and Great Britain in response to military action by the former Soviet Union: Soviet troopsclosed all roads and rail lines between West Germany and Berlin, cutting off supply routes to the city. The Allies used a mathematical technique developed during WorldWar II to maximize the amount of supplies transported. During the 15-month airlift, 278,228 flights provided basic necessities to blockaded Berlin, saving one of theworld's great cities.

In this section, we will look at an important application of systems of linear inequalities. Such systems arise in linear programming, a method for solving problems inwhich a particular quantity that must be maximized or minimized is limited by other factors. Linear programming is one of the most widely used tools in managementscience. It helps businesses allocate resources to manufacture products in a way that will maximize profit. Linear programming accounts for more than 50% andperhaps as much as 90% of all computing time used for management decisions in business. The Allies used linear programming to save Berlin.

Objective Functions in Linear Programming

1 Write an objective function describing a quantity that must be maximized or minimized.

Many problems involve quantities that must be maximized or minimized. Businesses are interested in maximizing profit. An operation in which bottled water and medicalkits are shipped to earthquake survivors needs to maximize the number of survivors helped by this shipment. An objective function is an algebraic expression in two ormore variables describing a quantity that must be maximized or minimized.

Example 1 Writing an Objective FunctionBottled water and medical supplies are to be shipped to survivors of an earthquake by plane. Each container of bottled water will serve ten people and each medicalkit will aid six people. If x represents the number of bottles of water to be shipped and y represents the number of medical kits, write the objective function thatdescribes the number of people who can be helped.



{3x + y < 9

3x + y > 9

{6x − y ≤ 24

6x − y > 24

{3x + y ≤ 9

3x + y ≥ 9

{6x − y ≤ 24

6x − y ≥ 24











Page 456







1/9/2018 thinking mathematically, sixth edition · 2018. 8. 31. · part of the solution set. thus,...

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