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.
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 3
Example Graphing a Linear
Inequality
Draw the graph of y £ 2x + 4. State the boundary of the region.
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 4
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.
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 5
Example Solving a System of
Inequalities Graphically
Solve the system 2x + 3y < 4 and y > x 2 .
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 6
Example Solving a System of
Inequalities Graphically
Solve the system 2x + 3y < 4 and y > x 2 .
Graph both inequalities and find their intersection.
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 7
Linear Programming
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 8
Example Solving a Linear
Programming Problem
Find the maximum and minimum values of the objective
function f = 3x + 4 y, subject to the constraints given by
the system of inequalities.
3x + 2 y £ 12
2x + 5y £ 19
x³0
y³0
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 9
Example Solving a Linear
Programming Problem
f = 3x + 4 y
3x + 2 y £ 12
2x + 5y £ 19
x³0
y³0
Copyright © 2011 Pearson, Inc.
The corner points are:
( 0, 0)
æ 19 ö
çè 0, 5 ÷ø
( 4,0)
( 2,3)
Slide 7.5 - 10
Example Solving a Linear
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).
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 11
Quick Review
Find the x- and y-intercepts of the line.
1. 3x + 4 y = 24
x
y
2.
+
=1
20 30
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
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 12
Quick Review Solutions
Find the x- and y-intercepts of the line.
1. 3x + 4 y = 24
(0,6) and (8,0)
x
y
2.
+
= 1 (0,30) and (20,0)
20 30
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
Copyright © 2011 Pearson, Inc.
(1.3,0.3)
Slide 7.5 - 13
Chapter Test
é -1 3ù
é 2 -1ù
1. Given A = ê
ú, B = ê
ú.
ë 4 0û
ë4 3 û
Find (a) A + B (b) A - B (c) - 2 A, and (d) 3A - 2B.
Find AB and BA, or state that a given product is not possible.
é -1 2 ù
é -2 3 1 ù
ê
ú
ê
ú
2. A = ê 3 -1ú , B = ê 2 1 0 ú
êë 4 3 úû
êë -1 2 -3úû
é 5 -3ù
3. A = éë -1 4 ùû , B = ê
ú
ë2 1 û
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 14
Chapter Test
4. Find the inverse matrix if it has one.
é -1 0 1ù
ê
ú
ê 2 -1 1ú
êë 1 1 1úû
5. Find the reduced row echelon form of the matrix
é 2 1 1 1ù
ê
ú
ê -3 -1 -2 1ú
êë 5 2 2 3úû
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 15
Chapter Test
6. Use Gaussian elimination to solve the system of equations.
x+z+w=2
x+ y+z =3
3x + 2 y + 3z + w = 8
7. Solve the system of equations by finding the reduced
row echelon form of the augmented matrix.
x + 2 y - 2z + w = 8
2x + 7 y - 7z + 2w = 25
x + 3y - 3z + w = 11
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 16
Chapter Test
3x - 2
8. Find the partial fraction decomposition of 2
.
x - 3x - 4
9.Find the minimum and maximum, if they exist, of the
objective function f , subject to the constraints.
Objective function: f = 7x + 6 y
Constraints: 7x + 5y ³ 100
2x + 5y ³ 50
x ³ 0, y ³ 0
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 17
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 of
shares 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.
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 18
Chapter Test Solutions
é -1 3ù
é 2 -1ù
é1 2 ù
1. Given A = ê
ú, B = ê
ú . Find (a) A + B ê
ú
4
0
4
3
8
3
ë
û
ë
û
ë
û
é -3 4 ù
é 2 -6 ù
é -7 11 ù
(b) A - B ê
ú (c) - 2 A, ê
ú (d) 3A - 2B. ê
ú
ë 0 -3û
ë -8 0 û
ë 4 -6 û
Find AB and BA, or state that a given product is not possible.
é -1 2 ù
é -2 3 1 ù
é15 -4 ù
ê
ú
ê
ú
ê
ú
2. A = ê 3 -1ú , B = ê 2 1 0 ú not possible; ê 1
3 ú
êë 4 3 úû
êë -1 2 -3úû
êë -5 -13úû
é 5 -3ù
3. A = éë -1 4 ùû , B = ê
ú éë3 7 ùû ; not possible
ë2 1 û
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 19
Chapter Test Solutions
4. Find the inverse matrix if it has one.
é -1 0 1ù
é -0.4 0.2 0.2 ù
ê
ú
ê
ú
ê 2 -1 1ú
ê -0.2 -0.4 0.6 ú
êë 1 1 1úû
êë 0.6 0.2 0.2 úû
5. Find the reduced row echelon form of the matrix
é 2 1 1 1ù
ê
ú
ê -3 -1 -2 1ú
êë 5 2 2 3úû
Copyright © 2011 Pearson, Inc.
é1 0 0 1 ù
ê
ú
ê0 1 0 2 ú
êë0 0 1 -3úû
Slide 7.5 - 20
Chapter Test Solutions
6. Use Gaussian elimination to solve the system of equations.
x+z+w=2
x+ y+z =3
3x + 2 y + 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 + 2 y - 2z + w = 8
2x + 7 y - 7z + 2w = 25
x + 3y - 3z + w = 11
Copyright © 2011 Pearson, Inc.
( -w + 2, z + 3, z, w)
Slide 7.5 - 21
Chapter Test Solutions
3x - 2
8. Find the partial fraction decomposition of 2
.
x - 3x - 4
1
2
+
x +1 x - 4
9. Find the minimum and maximum, if they exist, of the
objective function f , subject to the constraints.
Objective function: f = 7x + 6 y
Constraints: 7x + 5y ³ 100
2x + 5y ³ 50
x ³ 0, y ³ 0
minimum is 106 at (10,6); no maximum
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 22
Chapter Test Solutions
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. [200 400 600 250]
(b) Write a 1´ 4 matrix P representing the price per share of each
stock. [$80 $120 $200 $300]
(c) Write a matrix product that gives the total cost of the stocks that
the customer bought. NP = $259, 000
T
Copyright © 2011 Pearson, Inc.
Slide 7.5 - 23