Unit 7.5

7.5
Systems of
Inequalities in
Two Variables
Copyright © 2011 Pearson, Inc.

What you’ll learn about
 Graph of an Inequality
 Systems of Inequalities
 Linear Programming
… and why
Linear programming is used in business and
industry to maximize profits, minimize costs, and to
help management make decisions.
Copyright © 2011 Pearson, Inc. Slide 7.5 - 2

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
Draw the graph of y  2x  4. State the boundary of the region.
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.

Example Solving a System of
Inequalities Graphically
Solve the system 2x  3y  4 and y  x2.

Example Solving a System of
Inequalities Graphically
Solve the system 2x  3y  4 and y  x2.
Graph both inequalities and find their intersection.

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  12
2x  5y  19
x  0
y  0

Programming Problem
The corner points are:
0, 0
0,
19
5

 

 
4,0
2,3
f  3x  4y
3x  2y  12
2x  5y  19
x  0
y  0

Programming Problem
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.
x
20

y
30
 1
Find the point of intersection of the two lines.
3. x  y  3 and 2x  y  5
4. x  y  1 and y  3x 1
5. 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.
x
20

y
30
 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 3
4 0

 

 
,B 

2 1
4 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 2
3 1
4 3









,B 

2 3 1
2 1 0
1 2 3









3. A  1 4 

,B 

5 3
2 1

 
 

Chapter Test
4. Find the inverse matrix if it has one.

1 0 1
2 1 1
1 1 1









5. Find the reduced row echelon form of the matrix

2 1 1 1
3 1 2 1
5 2 2 3










Chapter Test
6. Use Gaussian elimination to solve the system of equations.
x  z  w  2
x  y  z  3
3x  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  8
2x  7 y  7z  2w  25
x  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  6y
Constraints: 7x  7 y  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 3
4 0

 

 
,B 

2 1
4 3

 
 . Find (a) A B
1 2
8 3

 

 
(b) A B

3 4
0 3

 
 
(c)  2A,
2 6
8 0

 

 
(d) 3A 2B.
7 11
4 6

 

 
Find AB and BA, or state that a given product is not possible.
2. A 

1 2
3 1
4 3









,B 

2 3 1
2 1 0
1 2 3









not possible;

15 4
1 3
5 13









3. A  1 4 

,B 

5 3
2 1

 
 
3 7 

; not possible

4. Find the inverse matrix if it has one.

1 0 1
2 1 1
1 1 1









0.4 0.2 0.2
0.2 0.4 0.6
0.6 0.2 0.2










5. Find the reduced row echelon form of the matrix

2 1 1 1
3 1 2 1
5 2 2 3










1 0 0 1
0 1 0 2
0 0 1 3










6. Use Gaussian elimination to solve the system of equations.
x  z  w  2
x  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  8
2x  7 y  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
.
1
x 1

2
x  4
9. Find the minimum and maximum, if they exist, of the
objective function f , subject to the constraints.
Objective function: f  7x  6y
Constraints: 7x  7 y  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
N
(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.
pectively.
200 400 600 250
P


 $80 $120 $200 $30
0

(c) Write a matrix product that gives the total cost of the stocks that
the customer bought.
NP T 
$259,000 Copyright © 2011 Pearson, Inc. Slide 7.5 - 23

Unit 7.5

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