1. 3.2 Linear Programming
3 Credits
AS 91574

2. Linear Programming I can plot linear inequalities
Learning objectives
I can plot linear inequalities
I can find regions of intersection

3. Inequalities and Regions
Finding the region for a single inequality
Shade the region for which x + 2y ≥ 6 x y 1 2 3 4 5 -1 -2 -3 -4 -5
(2,4)
2 + 2 x 4 = 10 ≥ 6 
Boundary line solid if inequality is either ≤ or ≥
1. Draw the boundary line equation x + 2y = 6.
x + 2y = 6  y = -½x + 3
y intercept 3, gradient –½
2. Choose a test point in one of the 2 regions to see whether or not it satisfies the inequality, then shade the required region.

4. Inequalities and Regions
Finding the region for a single Inequality x y 1 2 3 4 5 -1 -2 -3 -4 -5
Shade the region for which 2x - y < -1
2 x = 5 > -1 
1. Draw the boundary line equation 2x - y = -1
Boundary line dotted if inequality is either < or >
(3,1)
2x - y = -1  y = 2x + 1
y intercept 1, gradient 2
2. Choose a test point in one of the 2 regions to see whether or not it satisfies the inequality then shade the required region.

5. Inequalities and Regions
Finding the region for two inequalities x y 1 2 3 4 5 -1 -2 -3 -4 -5
Shade and label with the letter R, the region for which y ≥ 1 and x > 2. R
Draw boundary line y = 1
lightly shade the region for which y ≥ 1 isn’t true.
Boundary line solid since inequality is ≥
Draw boundary line x = 2
lightly shade the region for which x > 2 isn’t true
Boundary line dotted since inequality is >
Identify the blank region that satisfies both inequalities and label.

6. R Inequalities and Regions Finding the region for two inequalities yx y 1 2 3 4 5 -1 -2 -3 -4 -5
Shade and label with the letter R, the region for which x + y < -2 and x ≤ 1
Boundary line dotted since inequality is <
Boundary line solid since inequality is ≤
Draw line x + y = -2  y = -x – 2
y intercept -2 and gradient -1
lightly shade the region for which x + y < -2 isn’t true
The origin (0.0) makes a useful test point.
Draw line x = 1
lightly shade the region for which x ≤ 1 isn’t true R
Identify the blank region that satisfies both inequalities and label.

7. R Inequalities and Regions
Inequalities that enclose a region of the plane. x y 1 2 3 4 5 -1 -2 -3 -4 -5
Shade and label with the letter R, the region for which y ≥ -3 and x > 1 and 2x + y < 3
Identify the overlapping region that satisfies all 3 inequalities and label.
The origin (0.0) makes a useful test point.
Draw line y = -3
lightly shade the region for which y ≥ -3 isn’t true.
Draw line x = 1 R
lightly shade the region for which x > 1 isn’t true
Draw line 2x + y = 3  y = -2x + 3
lightly shade the region for which 2x + y < 3 isn’t true
y intercept 3 and gradient - 2

8. R Inequalities and Regions
Inequalities that enclose a region of the plane. x y 1 2 3 4 5 -1 -2 -3 -4 -5
Shade and label with the letter R, the region for which y ≥ -3, x > -2, y  2x - 3 and x + y < 2
Draw line y = -3
Draw line x + y = 2  y = -x + 2
The origin (0.0) makes a useful test point.
The origin (0.0) makes a useful test point.
lightly shade the region for which y ≥ -3 isn’t true
y intercept 2, gradient 1
lightly shade the region
x + y < 2 isn’t true
Draw line x = -2
lightly shade the region for which x > -2 isn’t true R
Identify the Blank region that satisfies all 4 inequalities and label.
Draw line y = 2x – 3
y intercept -3 and gradient 2.
lightly shade the region for which y  2x – 3 isn’t true

9. State clearly you are shading the region that does not satisfy the inequality

10. Success Criteria I can plot linear inequalities
I can find regions of intersection
Delta 3.01 page 47
3.02 page 48