Section 5.7
Systems of Inequalities
And Linear Programming
Linear Inequality in Two Variables
An inequality that can be written as
Ax + By < C or Ax + By > C,
where A, B, and C are real numbers
and A and B are not both 0.
The symbol < may be replaced with , >, or .
Linear Inequalities
The solution set of an inequality is theset of all ordered pairs that make it true.
The graph of an inequality representsits solution set.
Graphing an Inequality
1.Draw the boundary line
•Make the inequality an equation.
•Graph the equation.
> or < Solid line
> or < Dashed line
2.Choose a test point. (Any point not on the graph.)
•Substitute test point into original inequality.
3.Shade the appropriate region.
•Shade the region that includes the test point if it makesthe inequality true.
•If the test point does not make the inequality true, shadethe other side of the line.
Example
•Graph y > x 4.
•We begin by graphing the related equationy = x 4.
•We use a dashed line because the inequality symbolis >. This indicates that the line itself is not in thesolution set.
Example continued
•Determine whichhalf-planesatisfies theinequality bychoosing a testpoint.
To Graph a Linear Inequality:A Recap
•Replace the inequality symbol with an equals signand graph this related equation. If the inequalitysymbol is < or >, draw the line dashed. If theinequality symbol is or , draw the line solid.
•The graph consists of a half-plane on one side ofthe line and, if the line is solid, the line as well. Todetermine which half-plane to shade, test a pointnot on the line in the original inequality. If that pointis a solution, shade the half-plane containing thatpoint. If not, shade the opposite half-plane.
Example
•Graph: 4x + 2y 8
•Graph the relatedequation, using a solidline.
Example continued
Determine which half-plane to shade bychoosing a test point.
Example
•Graph x > 2 on a plane.
1. Graph the relatedequation.
2. Pick a test point (0, 0).
x > 2
0 > 2 False
Because (0, 0) is not asolution, we shade thehalf-plane that does notcontain that point.
Example
•Graph y 2 on a plane.
1. Graph the relatedequation.
2. Select a test point (0, 0).
y 2
0 2 True
Because (0, 0) is asolution, we shade theregion containing thatpoint.
Systems of Linear Inequalities
•Graph the solution set of thesystem.
•First, we graph x + y 3 usinga solid line.
Choose a test point (0, 0)and shade the correct plane.
•Next, we graph x y > 1 usinga dashed line.
Choose a test point andshade the correct plane.
The solution set of the system ofequations is the region shadedboth red and green, includingpart of the line x + y 3.
Example
•Graph the following system of inequalitiesand find the coordinates of any verticesformed:
Example continued
We graph the related equations using solid lines.
We shade the region common to all three solutionsets.
Example continued
The system of equations from inequalities (1) and (2):
y + 2 = 0
x + y = 2
The vertex is (4, 2).
The system of equations from inequalities (1) and (3):
y + 2 = 0
x + y = 0
The vertex is (2, 2).
The system of equations from inequalities (2) and (3):
x + y = 2
x + y = 0
The vertex is (1, 1).
To find the vertices, we solve three systems of equations.
Linear Programming
•In many applications, we want to find amaximum or minimum value. Linearprogramming can tell us how to do this.
•Constraints are expressed as inequalities.The solution set of the system of inequalitiesmade up of the constraints contains all thefeasible solutions of a linear programmingproblem.
•The function that we want to maximize orminimize is called the objective function.
Linear Programming Procedure
•To find the maximum or minimum value of alinear objective function subject to a set ofconstraints:
1.Set up objective function and define constraints.
2. Graph the region of feasible solutions.
3.Determine the coordinates of the vertices of theregion.
4.Evaluate the objective function at each vertex.The largest and smallest of those values are themaximum and minimum values of the function,respectively.
Example
•A tray of corn muffins requires 4 cups of milkand 3 cups of wheat flour.
A tray of pumpkin muffins requires 2 cups ofmilk and 3 cups of wheat flour.
There are 16 cups of milk and 15 cups of wheatflour available, and the baker makes $3 per trayprofit on corn muffins and $2 per tray profit onpumpkin muffins.
How many trays of each should the baker makein order to maximize profits?
Example continued
•A tray of corn muffins requires 4 cups of milk and 3 cups of wheatflour.
A tray of pumpkin muffins requires 2 cups of milk and 3 cups ofwheat flour.
There are 16 cups of milk and 15 cups of wheat flour available, andthe baker makes $3 per tray profit on corn muffins and $2 per trayprofit on pumpkin muffins.
Solution:
We let x = the number of corn muffins andy = the number of pumpkin muffins.
Then the profit P is given by the functionP = 3x + 2y.
Example continued
•A tray of corn muffins requires 4 cups of milk and 3 cups of wheat flour.
A tray of pumpkin muffins requires 2 cups of milk and 3 cups of wheat flour.
There are 16 cups of milk and 15 cups of wheat flour available, and the baker makes$3 per tray profit on corn muffins and $2 per tray profit on pumpkin muffins.
•We know that x muffins require 4 cups of milk and y muffinsrequire 2 cups of milk. Since there are no more than 16 cupsof milk, we have one constraint. 4x + 2y 16
•Similarly, the muffins require 3 and 3 cups of wheat flour.There are no more than 15 cups of flour available, so wehave a second constraint.
3x + 3y 15
•We also know x 0 and y 0 because the baker cannotmake a negative number of either muffin.
Example continued
•Thus we want to maximize the objectivefunction P = 3x + 2y
subject to the constraints:
4x + 2y 16,
3x + 3y 15,
x 0,
y 0.
We graph the system of inequalities anddetermine the vertices.
Next, we evaluate the objective function P ateach vertex.
Example continued
P = 3(3) + 2(2) = 13
(3, 2)
P = 3(0) + 2(5) = 10
(0, 5)
P = 3(4) + 2(0) = 12
(4, 0)
P = 3(0) + 2(0) = 0
(0, 0)
Profit P = 3x+ 2y
Vertices
Maximum
The baker will make a maximum profit when 3 traysof corn muffins and 2 trays of pumpkin muffins areproduced.
Example
•Omar owns a car and a moped. He can afford12 gal of gasoline to be split between the carand the moped. Omar’s car gets 20 mpg and,with the fuel currently in the tank, can hold atmost an additional 10 gal of gas. His mopedgets 100 mpg and can hold at most 3 gal of gas.
•How many gallons of gasoline should eachvehicle use if Omar wants to travel as far aspossible?
•What is the maximum number of miles that hecan travel?
Example continued
Omar owns a car and a moped. He can afford 12 gal of gasoline to besplit between the car and the moped. Omar’s car gets 20 mpg and,with the fuel currently in the tank, can hold at most an additional 10 galof gas. His moped gets 100 mpg and can hold at most 3 gal of gas.
Vertex
M = 20x + 100y