copyright © 2011 pearson, inc. 7.5 systems of inequalities in two variables

Copyright © 2011 Pearson, Inc.

7.5Systems of

Inequalities in Two Variables

5 - 2 Copyright © 2011 Pearson, Inc.

What you’ll learn about

Graph of an Inequality Systems of Inequalities Linear Programming

… and whyLinear programming is used in business and industry to maximize profits, minimize costs, and to help management make decisions.


Steps for Drawing the Graph of an Inequality in Two Variables

1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is < or>. Use a solid line if the inequality is ≤ or ≥.

2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.


Example Graphing a Linear Inequality

Draw the graph of y ≤2x+ 4. State the boundary of the region.


Example Graphing a Linear Inequality

Because of "≤," the graph of y=2x+ 4 is part of the

graph of the inequality. The point (0,0) satisfies the

inequality because 0 ≤2(0) + 4. Thus the graph of

y≤2x+ 4 consists

of all of the points

on or below the line

y=2x+ 4.

Draw the graph of y ≤2x+ 4. State the boundary of the region.


Example Solving a System of Inequalities Graphically

Solve the system 2x +3y< 4 and y> x2 .


Example Solving a System of Inequalities Graphically

Graph both inequalities and find their intersection.

Solve the system 2x +3y< 4 and y> x2 .


Linear Programming

Sometimes decision making in management science

requires that we find a minimum or a maximum of

a linear function

f =a1x1 + a2x2 +L + anxn

called an objective function, over a set of points.Such a problem is a linear programming problem.The feasible xy points or set of points is the solution

of inequalities, called constraints. The solution occurs

at one of the vertex points, or corner points,along the boundary region.


Example Solving a Linear Programming Problem

Find the maximum and minimum values of the objective

function f =3x+ 4y, subject to the constraints given by

the system of inequalities.3x+ 2y≤122x+5y≤19

x≥0y≥0



The corner points are:

0, 0( )

0,195

⎛

⎝⎜⎞

⎠⎟

4,0( )

2,3( )

f =3x+ 4y3x+ 2y≤122x+5y≤19x≥0y≥0



The following table evaluates f at the corner points

of the region.

The maximum value of f is 18 at (2, 3)

The minimum value is 0 at (0, 0).


Quick Review

Find the x- and y-intercepts of the line.

1. 3x + 4y=24

2. x20

+y30

=1

Find the point of intersection of the two lines.3. x+ y=3 and 2x−y=54. x−y=−1 and y=3x+15. 7x+3y=10 and x−y=1


Quick Review Solutions

Find the x- and y-intercepts of the line.

1. 3x + 4y=24 (0,6) and (8,0)

2. x20

+y30

=1 (0,30) and (20,0)

Find the point of intersection of the two lines.3. x+ y=3 and 2x−y=5 (8/3,1/3)4. x−y=−1 and y=3x+1 (0,1)5. 7x+3y=10 and x−y=1 (1.3,0.3)


Chapter Test

1. Given A =−1 34 0

⎡

⎣⎢

⎤

⎦⎥,B=

2 −14 3

⎡

⎣⎢

⎤

⎦⎥.

Find (a) A+ B (b) A−B (c) −2A, and (d) 3A−2B.Find AB and BA, or state that a given product is not possible.

2. A=−1 23 −14 3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,B=

−2 3 12 1 0−1 2 −3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3. A= −1 4⎡⎣ ⎤⎦,B=5 −32 1

⎡

⎣⎢

⎤

⎦⎥


Chapter Test

4. Find the inverse matrix if it has one.

−1 0 12 −1 11 1 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

5. Find the reduced row echelon form of the matrix

2 1 1 1−3 −1 −2 15 2 2 3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


Chapter Test

6. Use Gaussian elimination to solve the system of equations.

x + z+ w=2x+ y+ z=33x+ 2y+3z+ w=8

7. Solve the system of equations by finding the reduced

row echelon form of the augmented matrix.x+ 2y−2z+ w=82x+7y−7z+ 2w=25x+3y−3z+ w=11


Chapter Test

8. Find the partial fraction decomposition of 3x −2

x2 −3x−4.

9.Find the minimum and maximum, if they exist, of the

objective function f , subject to the constraints.Objective function: f =7x+6yConstraints: 7x+7y≥100

2x+5y≥50 x≥0, y≥0


Chapter Test

10. A stockbroker sold a customer 200 shares of stock A,

400 shares of stock B, 600 shares of stock C, and 250

shares of stock D. The price per share of A, B, C, and D

are $80, $120, $200, and $300, respectively.

(a) Write a 1×4 matrix N representing the number or

share of each stock the customer bought.(b) Write a 1×4 matrix P representing the price per

share of eachstock.(c) Write a matrix product that gives the total cost of the

stocks that the customer bought.


Chapter Test Solutions

1. Given A =−1 34 0

⎡

⎣⎢

⎤

⎦⎥,B=

2 −14 3

⎡

⎣⎢

⎤

⎦⎥. Find (a) A+ B

1 28 3

⎡

⎣⎢

⎤

⎦⎥

(b) A−B −3 40 −3

⎡

⎣⎢

⎤

⎦⎥ (c) −2A,

2 −6−8 0

⎡

⎣⎢

⎤

⎦⎥ (d) 3A−2B.

−7 114 −6

⎡

⎣⎢

⎤

⎦⎥

Find AB and BA, or state that a given product is not possible.

2. A=−1 23 −14 3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥,B=

−2 3 12 1 0−1 2 −3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ not possible;

15 −41 3−5 −13

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

3. A= −1 4⎡⎣ ⎤⎦,B=5 −32 1

⎡

⎣⎢

⎤

⎦⎥ 3 7⎡⎣ ⎤⎦; not possible



4. Find the inverse matrix if it has one.

−1 0 12 −1 11 1 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−0.4 0.2 0.2−0.2 −0.4 0.60.6 0.2 0.2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

5. Find the reduced row echelon form of the matrix

2 1 1 1−3 −1 −2 15 2 2 3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 0 0 10 1 0 20 0 1 −3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥



6. Use Gaussian elimination to solve the system of equations.

x + z+w=2x+ y+ z=3

3x+2y+3z+w=8 −z−w+2,w+1,z,w( )

7. Solve the system of equations by finding the reduced row echelon

form of the augmented matrix.x+2y−2z+w=82x+7y−7z+2w=25

x+3y−3z+w=11 −w+2,z+3,z,w( )



8. Find the partial fraction decomposition of 3x −2

x2 −3x−4.

1x+1

+2

x−49. Find the minimum and maximum, if they exist, of the

objective function f , subject to the constraints.Objective function: f =7x+6yConstraints: 7x+7y≥100

2x+5y≥50 x≥0, y≥0

minimum is 106 at (10,6); no maximum



10. A stockbroker sold a customer 200 shares of stock A, 400 shares

of stock B, 600 shares of stock C, and 250 shares of stock D. The

price per share of A, B, C, and D are $80, $120, $200, and $300,

res

[ ]

pectively.(a) Write a 1 4 matrix representing the number or share of each

stock the customer bought.

(b) Write a 1 4 matrix representing the price per share of each

stock.

200 400 600 250

$80 $120 $200 $30

N

P

×

×

[ ](c) Write a matrix product that gives the total cost of the stocks that

the customer bought.

0

$259, 0 00TNP =

copyright © 2011 pearson, inc. 7.5 systems of inequalities in two variables

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