1. Systems of Inequalities. Linear Programming
What you’ll learn
To solve systems of linear inequalities.
To solve real world problems involving systems
of linear inequalities.
To solve problems using linear programming.
Vocabulary
Constrain, linear programming,
feasible region,
objective function.

2. Take a note An inequality and a system of inequalities can each have
many solutions. A solution of a system of inequalities is a
solution for each inequality of the system.
You can solve a system of inequalities in more then one way. Graphing the solution is usually the most appropriate method. The solution is the set of all points that are solutions of each inequality in the system.
When solving a system of linear inequalities by graphing.
A graphed solution consist of a half a plane and possibly
a boundary line ,so the solution is the overlap of the
two half planes.

3. This is the solution for the system
Solving a system by graphing.
What is the solution of the system of inequalities?
This is the solution for the system

4. What is the solution of the system of inequalities?
Your turn
What is the solution of the system of inequalities?
Solution of
the system
y ≥x-3
y-3x≤-4
y ≥x-3
Answer:
y-3x≤-4

5. Your turn again
Solving a linear/absolute value system
What is the solution of the system of inequalities? x y

6. and the inside is called feasible region
Linear programming
Constrains (borders)
and the inside is called feasible region
Graph
X=0 then y=10 (0,10)
y=0 then x=10 (10,0)
What is the solution for all inequalities at once?

7. If there is a maximum or a minimum value of the linear
Take a note
The constrains in a linear programming
Situation form a system of inequalities,
like the one you did before. The graph of the system
(the purple part) is the feasible region. It contains all
the points that satisfy all the constrains.
The quantity you are trying to maximize or minimize is
modeled with an objective function. Often this quantity is
cost or profit.
If there is a maximum or a minimum value of the linear
objective function, it occurs at one or more vertices
of the feasible region.
Some real world problems involved multiple linear
relationships. Linear programming accounts for all these
linear relationships and give the solution to the problem.

8. Testing vertices. Multiple choice
Linear programming
Testing vertices. Multiple choice
What point in the feasible region maximizes P for
the objective function P=2x+y
Answer:
A.(2,0)
B.(0,0)
C.(3,1)
D.(0,2.5)
Step 1: graph the inequalities.
Step 2: form the feasible region
Step 3: find the coordinates of each vertex
Step 4: evaluate P at each vertex.

9. x=0 then y=-2 and y=0 then x=2
Coordinates of each vertex
First Q.
A(2,0)
B(0,0)
C(3,1)
D(0,2.5)
Evaluate
P=2(0)+0=0
P=2(0)+2.5=2.5
P=2(3)+1=7
P=2(2)+0=4 C
Maximum
value D B A
Answer: C

10. Classwork odd Homework even