1. Systems of Linear Inequalities with Three Inequalities
AII, 2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

2. Systems of Linear Inequalities
Objective
Key Words
Graph, write, and use a system of linear inequalities.
Three Inequalities
Why we need this? It will prepare you for college in related fields such as math and science.
System of linear inequalities in two variables
Think about what a system of equations looks like
Solution of a system of linear inequalities
Think about shaded regions
Graph of a system of linear inequalities
Think about dashed lines, solid lines, and shaded regions

3. Pre-requisite Check Graph system of inequalities y<-4x x+3y≥3
Check Solutions of Inequalities
y<-4x
x+3y≥3
Check whether (4,2) is a solution of the system of inequalities.
𝑥+𝑦>−2 𝑥−𝑦≥4
𝑥−3𝑦≤4 𝑥>1

4. You-Try Exercises
From Yesterday!
Graph and test a solution point (ie. (0,0)) ≤ y x +
ANSWER

5. You-Try Exercises
From Yesterday!
Check whether (3, 1) is a solution of the system of
inequalities. –
< y 2 1
> x + 1.
ANSWER
solution 2. 4 y – 2x + ≤ 6 2y
> x
ANSWER
not a solution 3. 4 y x – ≥ 8 4y
> 3x +
ANSWER
not a solution

6. Graphing a System of Linear Inequalities
Step 1:
Graph the boundary lines of the inequalities. Use a dashed line for an inequality with < or >. Use a solid line for an inequality with ≤ or ≥.
Step 2:
Shade the half-planes for the inequalities. The graph of the system is the region common to all the half-planes.
𝑦≥−1 𝑥<2 𝑦>𝑥

7. YOU DO: Graph the system.
Checkpoint
Graph a System of Two Inequalities
From Yesterday!
YOU DO: Graph the system. 3. + y x 4
> y 2x 3 ≥ –
ANSWER

8. The inequality implies that the region is on and above the line ≥ y 2
Example 1
Graph a System of Three Inequalities
I DO: Graph the system. ≥ x 2
Inequality 1 ≥ y 2 –
Inequality 2 ≤
3x y 12 +
Inequality 3
SOLUTION
The inequality implies that the region is on and to the right of the line ≥ x 2 = .
The inequality implies that the region is on and above the line ≥ y 2 – = .

9. The inequality implies that the region is on and below the line . ≤
Example 1
Graph a System of Three Inequalities
The inequality implies that the region is on and below the line ≤
3x y 12 + =
3x y 12 +
The graph of the system is the shaded triangular region shown at the bottom. 9

10. I DO #4 and YOU DO #5: Graph the system.
Checkpoint
Graph a System of Three Inequalities
I DO #4 and YOU DO #5: Graph the system. 4. ≥ x y
x y 5 +
<
ANSWER 5. ≥ x y 2 2x +
< 6
ANSWER

11. YOU DO: Graph the system.
Checkpoint
Graph a System of Three Inequalities
YOU DO: Graph the system. 6. ≤ y 5
< x 2 3x + ≥
ANSWER

12. Conclusion Summary Assignment
How do you find the solution of a system of linear inequalities?
Graph each inequality. The solution is the overlapping shaded region, if any.
To be completed in class
Pg 188 #(20,26,42,54)