1. Do Now! Solve the system of equations Do all work on the notecard.

2. Linear Inequalities and Linear Programming
Day 1 – Graphing Inequalities

3. Inequality Symbol Type of Line > or < Dotted Line Solid Line
How to determine what type of line to draw:
Inequality Symbol
Type of Line
> or <
Dotted Line
Solid Line

4. Choose the type of line for the inequality given.
1. y > 3x - 2
a. Solid b. Dotted
2. y > ¼x - 5

5. Choose the inequality symbol for the line shown.

6. Choose the inequality symbol for the line shown.

7. How to Shade: If the inequality is: Shade y > mx + b or
Above the line
y < mx + b
Below the line

8. How to Shade:
A good tip is to test a point on one side of the line. A good point to test is (0,0)
Example:
y > 3x plug in 0 for x and y
0 > 3(0) simplify
0 > is it true??
No, then shade on the shade NOT containing the point (0,0)

9. Graph y > x - 2. 1. Graph the line y = x - 2. x
2. Since y >, shade above the line. y

10. Graph y < x - 2. 1. Graph the line y = x - 2. x
2. Since y <, shade below the line. y

11. Graph y > x - 2. 1. Graph the line
y = x - 2, but make the line dotted. x
2. Since y >, shade above the line. y

12. Graph y < x - 2. 1. Graph the line
y = x - 2, but make the line dotted. x
2. Since y <, shade below the line. y

13. Graph y > -½x + 3
Type of line:
Solid
Dotted x y

14. Graph y > -½x + 3 Type of line: Solid Dotted x Shade ___ the line.
Above
Below

15. Graph y > -½x + 3 Type of line: Solid Dotted x Shade ___ the line.
Above
Below

16. Choose the correct inequality for the graph shown.
y < 1/3 x + 2
y < 1/3 x + 2 x
y > 1/3 x + 2
y > 1/3 x + 2 y

17. Graph x > -2 1. Draw a dotted vertical line at x = -2.
2. Test the point (0,0) x
3. Shade to the right of the line. y

18. Graph x > 3.
Choose type of line.
Solid
Dotted x y

19. Graph x > 3. Choose type of line. Solid Choose where to shade. Left
Right y

20. Graph x > 3. Choose type of line. Solid Choose where to shade.
Right y

21. Solve -3x - 2y < 12.
+3x +3x
-2y < 3x + 12
y < -3/2 x - 6
>

22. Choose the correct inequality. 1. 2x + 5y > -10

23. Example 1 Which ordered pair is a solution of 5x - 2y ≤ 6? (0, -3)
(5, 5)
(1, -2)
(3, 3)
ANSWER: A. (0, -3)

24. Example 2 Graph the inequality x ≤ 4 in a coordinate plane.
Decide whether to
use a solid or
dashed line.
Use (0, 0) as a
test point.
Shade where the
solutions will be. y x 5 -5

25. Example 3 Graph 3x - 4y > 12 in a coordinate plane.
Sketch the boundary line of the graph.
Solve for “y” first:
y < ¾x - 3
Solid or dashed
line?
Use (0, 0) as a
test point.
Shade where the
solutions are. y x 5 -5

26. Graph y < 2/5x Graph y < 2/5x in a coordinate plane.
What is the slope and y-intercept?
m = 2/5
b = (0,0)
Solid or dashed
line?
Use a test point
OTHER than the
origin.
Shade where the
solutions are. y x 5 -5

27. Graph: y ≥ -3/2x + 1 Step 1: graph the boundary (the line is solid ≥)
Step 2: test
a point NOT
On the line
(0,0) is always
The easiest if it’s
Not on the line!!
3(0) + 2(0) ≥ 2
0 ≥ 2
Not a solution
So shade the other side of the line!!

28. Graph: y < 6

29. Graph: 4x – 2y < 7

30. Linear Inequalities and Linear Programming
Day 2 - Word Problems

31. Warm Up
1. Graph the system of inequalities and classify the figure created by the solution region.
y ≤ x – 2
y ≥ –2x – 2
x ≤ 4
x ≥ 1

32. Objective
Solve linear programming problems.

33. Vocabulary linear programming constraint feasible region
objective function
vertex

34. Linear programming is method of finding a maximum or minimum value of a function that satisfies a given set of conditions called constraints. A constraint is one of the inequalities in a linear programming problem. The solution to the set of constraints can be graphed as a feasible region.

35. In most linear programming problems, you want to do more than identify the feasible region. Often you want to find the best combination of values in order to minimize or maximize a certain function. This function is the objective function.
The objective function may have a minimum, a maximum, neither, or both depending on the feasible region.

36. More advanced mathematics can prove that the maximum or minimum value of the objective function will always occur at a vertex of the feasible region.

37. Example 1
Graph each feasible region
Maximize the objective function P = 25x + 30y under the following constraints.
x ≥ 0
y ≥ 1.5
2.5x + 5y ≤ 20
3x + 2y ≤ 12

38. Example 1 Continued
A. Use the constraints to graph the feasible region.
x ≥ 0
y ≥ 1.5
2.5x + 5y ≤ 20
3x + 2y ≤ 12

39. Example 1 Continued
B. Evaluate the objective function at the vertices of the feasible region.
(x, y)
25x + 30y
P($)
(0, 4)
25(0) + 30(4)
120
(0, 1.5)
25(0) + 30(1.5) 45
(2, 3)
25(2) + 30(3)
140
(3, 1.5)
25(3) + 30(1.5)
The maximum value occurs at the vertex (2, 3).
P = 140

40. Example 2: Graphing a Feasible Region
Yum’s Bakery bakes two breads, A and B. One batch of A uses 5 pounds of oats and 3 pounds of flour. One batch of B uses 2 pounds of oats and 3 pounds of flour. The company has 180 pounds of oats and 135 pounds of flour available. Write the constraints for the problem and graph the feasible region.

41. Let x = the number of bread A, and y = the number of bread B.
Example 2 Continued
Let x = the number of bread A, and
y = the number of bread B.
Write the constraints:
x ≥ 0
The number of batches cannot be negative.
y ≥ 0
The combined amount of oats is less than or equal to 180 pounds.
5x + 2y ≤ 180
The combined amount of flour is less than or equal to 135 pounds.
3x + 3y ≤ 135

42. Graph the feasible region
Graph the feasible region. The feasible region is a quadrilateral with vertices at (0, 0), (36, 0), (30, 15), and (0, 45).
Check A point in the feasible region, such as (10, 10), satisfies all of the constraints. 

43. Example 3: Solving Linear Programming Problems
Yum’s Bakery wants to maximize its profits from bread sales. One batch of A yields a profit of $40. One batch of B yields a profit of $30. Use the profit information and the data from Example 1 to find how many batches of each bread the bakery should bake.

44. Example 3 Continued
Step 1 Let P = the profit from the bread.
Write the objective function: P = 40x + 30y
Step 2 Recall the constraints and the graph from Example 1.
x ≥ 0
y ≥ 0
5x + 2y ≤ 180
3x + 3y ≤ 135

45. Example 3 Continued
Step 3 Evaluate the objective function at the vertices of the feasible region.
(x, y)
40x + 30y
P($)
(0, 0)
40(0) + 30(0)
(0, 45)
40(0) + 30(45)
1350
(30, 15)
40(30) + 30(15)
1650
(36, 0)
40(36) + 30(0)
1440
The maximum value occurs at the vertex (30, 15).
Yum’s Bakery should make 30 batches of bread A and 15 batches of bread B to maximize the amount of profit.

46. Try it on your own!
1. Ace Guitars produces acoustic and electric guitars. Each acoustic guitar yields a profit of $30, and requires 2 work hours in factory A and 4 work hours in factory B. Each electric guitar yields a profit of $50 and requires 4 work hours in factory A and 3 work hours in factory B. Each factory operates for at most 10 hours each day. Graph the feasible region. Then, find the number of each type of guitar that should be produced each day to maximize the company’s profits.

47. Try it on your own!
Constraints:
Objective Function:
30x + 50y
x ≥ 0
y ≥ 0
2x + 4y ≤ 10
4x + 3y ≤ 10
1 acoustic; 2 electric