Abstract
This lesson is designed to discuss concepts of Linear Inequalities as use them to maximize profit in a business they
create.
Standards (NCTM)
Algebra
Represent and analyze mathematical situations and structures using algebraic symbols.
 Develop an initial conceptual understanding of different uses of variables;
 Explore relationships between symbolic expressions and graphs of lines, paying particular
attention to the meaning of intercept and slope.
 Use symbolic algebra to represent situations and to solve problems, especially those that involve
linear relationships;
 Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations.
Use mathematical models to represent and understand quantitative relationships.
 Model and solve contextualized problems using various representations, such as graphs, tables and
equations.
Student Prerequisites
 Arithmetic:
Students must be able to:
 perform basic operations with numbers such as addition, subtraction, multiplication, and
division
 Algebra:
Students must be able to:
 solve for a variable in linear equations.
 Technological:
Students must be able to:
 perform basic mouse manipulations such as point, click, and drag.
 use a browser such as Safari for experimenting with the activities.
Teacher Preparation
Teacher will need:
 Article The Walmart You Don't Know (Fast Company, Issue 77 December 2003 by Charles Fishman)
or another article that relates pricing with profit.
Students will need:
 access to a browser.
 pencil and paper.
Lesson Outline
 Focus and Review
 Read the portions of the article that pertain to Vlasic pickles.
 Explain that this article exemplifies how pricing a product affects profit and reminds consumers that the
objective of a business is to maximize profit.
 Objectives
Students will work with and solve linear inequalities in two variables using a combination of algebra and graphing.
Students will also get to see a practical application of linear inequalities by creating a business and maximizing
their profit.
 Guided Practice
 Ask the students if they have seen a problem like 10x+5y &le 25. Ask them if they know how to solve this
algebraically and graphically. Have students solve this problem on their own and then have one of them demonstrate
their solutions in front of the class.
 Ask students if they know how to solve a system of linear inequalities like:
10x+5y &le 25
12x+3y &le 27
 What does it mean to solve a system of linear inequalities? How would you graph these? Have the students graph these on their
own using graph paper.
 Teacher Input
 Make sure the students understand that solving a system of linear inequalities means to find the graph of all
ordered pairs of real numbers that simultaneously satisfy all the inequalities in the system.
 Using a graphing program on your computer, preferably one that graphs inequalities, graph this
set of inequalities. If you don't have such a program, use Graph Sketcher
on Interactivate's website.
 Make sure a few definitions and concepts are made clear:
 Solution or feasible region: the graph of all the ordered pairs of real numbers that
satisfies all the linear inequalities simultaneously.
 Corner Points: a point in the solution region that is the intersection of two boundary lines.
 Explain that we are going to use this knowledge of Linear Inequalities to talk about Linear Programming.
Linear Programming: a mathematical process that has been developed to help management in decision making.
 Describe the activity: Using Linear Programming, students are going to create a business where the purpose
is to maximize profit!! They will design the products and determine the labor departments, hours, and profit.
This is a challenging activity so go over an example so students see how
everything is played out.
 In your business, you decide to produce M&Ms and Snicker Bars. Your two labor departments for production
are the Chocolate Coating department and the Packaging Department. It takes .7 hours to chocolate coat the M&Ms
and 1.2 hours to package them. For Snickers, it takes 1.1 hours to chocolate coat and .8 hours to package. With
this information, develop two linear equalities if the maximum labor hours for the chocolate coating and packaging
departments are 500 and 456 hours.
 If M&Ms are sold for a $1.00 a package, your profit might be about $.30 and if the Snickers are sold at
$.75, your profit would be about $.25 (these numbers are made up). Thus, your profit equation would be:
P=.30X+.25Y Once again, the objective is to maximize profit!
 Steps to follow using your example:
 Determine your variables. X: # of M&Ms produced monthly. Y: # of Snickers produced monthly.
 Determine Labor Departments: Chocolate Coating and Packaging
 Determine Labor Hours: .7 hours for M&Ms and 1.1 hours for Snickers in the Chocolate Coating Department;
1.2 hours for M&Ms and .8 hours for Snickers in the Packaging Department. We also decided that the hours
restrictions per month were 500 and 456 hours for the two departments.
 Write inequalities and other restrictions:
.7X+1.1Y &le 500
1.2X+.8Y &le 456
X &ge 0
Y &ge 0
 Write Profit Equation: P=.30X+.25Y (For every M&Ms sold, you make $.30 and for every Snickers sold you make
$.25).
 Graph the inequalities using Graph Sketcher
and determine the feasible region.
 Find the Corner Points:
Fundamental Theorem of Linear Programming:If the optimal value of the objective function
in a linear programming problem exists, then the value must occur at one (or more) of the corner points
of feasible region. Therefore, the optimization (maximum profit) will occur at one of the corner points.
 Plug corner points into Profit equation (you should have three) and determine which pair maximizes profit
(this is under the assumption that all the M&Ms and Snickers that you produce are sold).
 Guided Practice
 Have the students, in pairs, create their own business modeling it similar to yours. Let them choose what products they will
produce, labor departments, labor hours, and the profit equation. They will want to use Graph Sketcher
to graph their inequalities and a mixture of both the graphs and algebra to solve for the corner points. Although the students
don't know exact values for labor hours and profit, let them come up with their own as the point of the assignment is to learn how
to use linear inequalities to maximize profit.
 Teacher Input
 Because students are developing their own equations, their lines may not intersect on the graph. Therefore, help them develop,
if necessary, equations that when graphed, intersect each other in the first quadrant.
 Guided Practice
 When students are done, have them share their results with the class, including the graph they found and how much of each product
they should sell to maximize profit.
 Closure
 Review the definitions, concepts, and procedures discussed in class. Ask them to share any thing they found was interesting/frustrating
and discuss their thoughts.

The Shodor Education Foundation, Inc.
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