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SYSTEMS OF INEQUALITIES

Unit Overview
In this unit, you are going to expand on the idea of systems of equations and learn how to graph linear inequalities and systems of linear inequalities.  Systems of linear inequalities can be used to establish ranges of possibilities for real-world situations, such as budgeting and cost.


Linear Inequalities



Quicktime_Video_Icon  Two Lines (11:59)




Graphing Systems of Linear Inequalities


Quicktime_Video_Icon  The Intersection (03:58)


Applications of Systems of Equations

When using systems of inequalities in the real world, this is often referred to as Linear Programming.  The concept is the same as above.  Write and graph each inequality (these are sometimes referred to as constraints) and the solutions to the problem will lie within the shaded region.   

Interestingly – when seeking a maximum or minimum value –this will always be one of the vertices of the shaded polygon within the bounded areas.  Let’s try an application: 

Maleigha is selling pizzas to raise money for prom.  Cheese pizzas cost $10 and pepperoni pizzas cost $12.  She must sell at least 5 of each kind, and she wants to sell at least $150 worth of pizzas.  Write linear inequalities based on the number of pizzas and the value of the pizzas, and then graph.

First:  Determine the variables that will be used to represent quantities of the different pizzas presented in the problem.   

Let x be the number of cheese pizzas.  Let y be the number of pepperoni pizzas.

Next:  Write inequalities based on the the number of pizzas and graph them.

Use ≥ to represent "at least."
 
She must sell at least 5 of each kind...

represents "at least 5 cheese pizzas"
 

Click here to view the graph.

represents "at least 5 cheese pizzas"
 

Click here to view the graph.

Next:  Write an inequality based on the value of the pizzas and graph it.

Cheese pizzas cost $10... 
 
10x  represents the value of the cheese pizzas
 
...  and pepperoni pizzas cost $12
 
12y  represents the value of the pepparoni pizzas
 
...  and she wants to sell at least $150 worth of pizzas.
 

 Put all of the values together. 

 
Click here to view the graph.

Finally:  Graph the inequalities in one plane.

Click here to view the graph.

Determine which of the following scenarios would be correct based on the graph of the inequalities and shows that Maleigha met her goal.

question1Maleigha should sell 3 cheese pizzas and 7 pepperoni pizzas.

No, this ordered pair (3, 7) is outside of the shaded region.

"Click here" to check the answer.


"Click here" to view the graph.


question1Maleigha should sell 10 cheese pizzas and 7 pepperoni pizzas.

Yes, this ordered pair (10, 7) is in the shaded region.

"Click here" to check the answer.


"Click here" to view the graph.


question1Maleigha should sell 5 cheese pizzas and 11 pepperoni pizzas.

Yes, this ordered pair (5, 11) is on the edge of the shaded region.

"Click here" to check the answer.


"Click here" to view the graph.

Quicktime_Video_Icon  Solving Systems of Inequalities (05:05)