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SECTION 3.4
Systems of Linear Inequalities
Warm Up
1. Graph y < -x +1 2. Graph the system.
{ 𝑦=𝑥+2𝑦=−2 𝑥−1
Systems of Linear Inequalities
• A system of linear inequalities is a collection of linear inequalities in the same variables.
• The solution is any ordered pair that satisfies each of the inequalities in the system. (anything in the shaded region.
• Each of the linear inequalities is called a boundary line for the solution.
Systems of Linear Inequalities
• Graph the system:
{ 𝑦 ≥1𝑥<−3
Systems of Linear Inequalities
• Graph the system:
{𝑦 ≤2𝑥𝑦>5𝑥≥3
Systems of Linear Inequalities
• Graph the system:
{𝑦 ≥−𝑥−1𝑦 ≤2𝑥+1
𝑥<1
Systems of Linear Inequalities
• Graph the system:
{ 𝑥 ≥0𝑦 ≥0
𝑦>−2 𝑥+5𝑦 ≤3𝑥+1
Systems of Linear Inequalities
• Graph the system:
{ 𝑥≥0𝑦 ≥0
𝑦<−𝑥+2𝑦 ≥−2 𝑥+3
Systems of Linear Inequalities
• Graph the system:
{𝑦 ≤ 12 𝑥−1
𝑦>−𝑥+2𝑦 ≥𝑥
Systems of Linear Inequalities
Write the system of inequalities graphed at right.
1. Find equations for the boundary lines.
2. Give each boundary line the appropriate inequality symbol.
A (0,4)
B (3, 5)
C (6, 0)
Linear Programing
• A method called linear programming is used to find optimal solutions such as the maximum revenue.
• Linear-programming problems have the following characteristics:• The inequalities contained in the problem are
called constraints.• The solution to the set of constraints is called the
feasible region.• The function to be maximized or minimized is
called the objective function.
Linear Programming: Example 1
Max Desmond is a farmer who plants corn and wheat. In making planting decisions, he used the 2010 statistics from the U.S. Bureau of Census below:
Mr. Desmond wants to plan according to the following constraints:• No more than 120 acres of corn and wheat• At least 20 and no more than 80 acres of corn• At least 30 acres of wheat
Crop Yield per acre
Average Price
Corn 113.5 bu $3.15/bu
Wheat 35.8 bu $4.45/bu
Linear Programming: Example 1
Mr. Desmond wants to plan according to the following constraints:• No more than 120 acres of corn and wheat• At least 20 and no more than 80 acres of corn• At least 30 acres of wheat
1. Write a system of inequalities to represent the constraints.
Linear Programming: Example 1
2. Graph the feasible region.
Linear Programming: Example 1
3. Write an objective function for the revenue from Mr. Desmond’s harvest.
R = (yield per acre)(average price)x + (yield per acre)(average price)y
Crop Yield per acre
Average Price
Corn 113.5 bu $3.15/bu
Wheat
35.8 bu $4.45/bu
Linear Programming1. Copy and complete the table to find the revenue at each of the four vertices of
the feasible region from Example 1.
2. Which vertex represents the greatest revenue?3. What do the coordinates of the vertex represent?4. Guess and check: Choose points on the boundary lines of the feasible region.
Find the corresponding revenues for these points. Can you find a point that gives a greater revenue than the vertex you chose in Step 2?
5. Guess and check: Choose points inside the feasible region. Find the corresponding revenues for these points. Can you find a point that gives a greater revenue than the vertex you chose in Step 2?
6. Do your investigations suggest that the maximum value of the objective function occurs at a vertex? Justify your response.
7. Look for a pattern: Repeat steps 2-6 for the minimum revenue instead of the max revenue. Explain how the points that correspond to the maximum and minimum revenues are related.
Vertex Objective function
A (20, 30) R = 357.525(20) + 159.31(30)
B (20, 100)
C (80, 40)
D (80, 30)
Linear Programming
Corner-Point Principle
In linear programming, the maximum and minimum values of the objective function each occur at one of the vertices of the feasible region.
Linear Programming: Example 2Find the maximum and minimum values, if they exist, of C = 3x +4y for each set of constraints:1.
Finding Corner-Points using your calculator
1. Put as many of the inequalities in slope-intercept form
2. Graph inequalities as equations in calculator3. Find the intersections of all lines drawn on picture4. Be sure to go back and take into consideration any
equation(s) that could not be put in slope-intercept form
• i.e. x <23 graphed as x = 235. Write down all of the vertices as you find them.6. Plug each vertex into the Objective Function to find
the maximum and/or minimum.
Linear Programming: Example 3Find the maximum and minimum values, if they exist, of C = 3x +4y for each set of constraints:1.
