chapter 5 systems of equations and inequalities copyright © 2014, 2010, 2007 pearson education,...

Chapter 5Systems of Equations

and Inequalities

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

5.5 Systems of Inequalities


• Graph a linear inequality in two variables.• Graph a nonlinear inequality in two variables.• Use mathematical models involving linear inequalities.• Graph a system of inequalities.

Objectives:


Linear Inequalities in Two Variables and Their Solutions

Equations in the form Ax + By = C are straight lines when graphed. If we change the symbol = to we obtain a linear inequality in two variables.

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 the y-coordinate is substituted for y in the inequality, we obtain a true statement. Each ordered-pair solution is said to satisfy the inequality.

, , , ,


The Graph of a Linear Inequality in Two Variables

The graph of an inequality in two variables is the set of all points whose coordinates satisfy the inequality.


Graphing a Linear Inequality in Two Variables


Example: Graphing a Linear Inequality in Two Variables

Graph:

Step 1 Replace the inequality symbol by = and graph the linear equation.

4 2 8x y

4 2 8x y We set y = 0 to find the x-intercept.

We set x = 0 to find the y-intercept.

4 2 8x y

4 2 8x y

4 2 0 8x 4 8x 2x

4 0 2 8y 2 8y 4y


Example: Graphing a Linear Inequality in Two Variables (continued)

Graph:

Step 1 (cont) Replace the inequality symbol by = and graph the linear equation.

We have found that the

x-intercept is 2

and the y-intercept is – 4.

Using the intercepts,

we graph the line.

4 2 8x y

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

4 2 8x y



Graph:

Step 2 Choose a test point from one of the half-planes and not from the line. Substitute its coordinates into the inequality.

We will test (0, 0).

4 2 8x y

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

4 2 8x y 4 0 2 0 8?

0 0 8? 0 8false



Graph:

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

The inequality is false

at (0, 0). We shade

the half-plane that

does not include (0, 0).

4 2 8x y

-4 -3 -2 -1 1 2 3 4

-5

-4

-3

-2

-1

1

2

3

4

5

x

y


Example: Graphing a Nonlinear Inequality in Two Variables

Graph:

Step 1 Replace the inequality symbol by = and graph the nonlinear equation.

The graph is a circle

of radius 4 with its

center at the origin.

2 2 16x y

2 2 16x y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y


Example: Graphing a Nonlinear Inequality in Two Variables (continued)

Graph:

Step 2 Choose a test point from one of the half-planes and not from the circle. Substitute its coordinates into the inequality.

We will test (0, 0).

2 2 16x y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

2 2 16x y 2 20 0 9? 0 0 9?

0 9

false


Example: Graphing a Nonlinear Inequality in Two Variables (continued)

Graph:

Step 3 If a true statement results, shade the region containing the test point.

The false statement

tells us that no points

inside the circle satisfy

the inequality. We shade

outside the circle.

2 2 16x y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y


Example: Application

The latest guidelines, which apply to both men and woman, give healthy weight ranges, rather than specific weights, for your height. 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:

Show that the point (66, 130) is a solution of this system.

4.9 165

3.7 125

x y

x y


Example: Application (continued)

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:

4.9 165

3.7 125

x y

x y

Show that the point (66, 130) is asolution of this system.

4.9 165x y 4.9 66 130 165?

323.4 130 165? 193.4 165

true

3.7 125x y 3.7 66 130 125?

244.2 130 125? 114.2 125

true


Example: Application (continued)

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:

The point (66, 130) satisfies both inequalities. Thus, (66, 130) is a solution for this system.

4.9 165

3.7 125

x y

x y

Show that the point (66, 130) is asolution of this system.


Graphing Systems 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 of inequalities 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 common to every graph in the system. This region of intersection gives a picture of the system’s solution set.


Example: Graphing a System of Linear Inequalities

Graph the solution set of the system:

Step 1 Replace the inequality symbol by = and graph the linear equations.

3 6

2 3 6

x y

x y

Find the intercepts.

3 6x y 2 3 6x y

Find the intercepts.3 0 6x

6x 0 3 6y

2y

2 3 0 6x 3x

2 0 3 6y 2y


Example: Graphing a System of Linear Inequalities (continued)


Step 1 (cont) Replace the inequality

symbol by = and graph

the linear equations.

The intercepts for

are (6, 0) and (0, –2).

The intercepts for

are (–3, 0) and (0, –2).

3 6

2 3 6

x y

x y

3 6x y

2 3 6x y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

x

y




Step 2 We will test the

half-planes formed by

each line.

3 6

2 3 6

x y

x y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

x

y

3 6x y 0 3 0 6? 0 6

(0,0) (0,0)2 3 6x y

2 0 3 0 6? 0 6

(0,0) is a solution for bothinequalities.




Step 3 We shade the

solution region.

3 6

2 3 6

x y

x y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

x

y

chapter 5 systems of equations and inequalities copyright © 2014, 2010, 2007 pearson education,...

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