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Graphing Systems of Inequalities
 A system of inequalities is two inequalities in the same two variables. The solution is the set of all ordered pairs that satisfy both inequalities.
 To solve a system of inequalities, graph each inequality. The solution of the system is the intersection of the graphs of the inequalities.
 If the intersection is empty, the system has no solution.
 In certain real world problems, only solutions that are positive numbers or integers make sense.
Graphing Systems of Inequalities
Solve:
y ≤ x
y <− x + 2
y ≤ x
y <− x + 2
 Graph y ≤ x
b = 0m = 1
Use test point (1.0)  Graph y <− x + 2b = 2m = − 1
Use test point (0,0)
Graph both together to find the area of overlap
Solve:
y > x y <− 2x
y > x y <− 2x
 Graph y > xb = 0m = 1
Use test point (1.0)  Graph y <− 2xb = 0m = − 2
Use test point (1,0)
Graph both together to find the area of overlap
Solve:
y > 3x + 1
y ≥ x − 1
y > 3x + 1
y ≥ x − 1
 Graph y > 3x + 1b = 1m = 3
Use test point (0.0)  Graph y ≥ x − 1b = − 1m = 1
Use test point (1,0)
Graph both together to find the area of overlap
Solve:
y ≤ 3
x ≥ 4
y ≤ 3
x ≥ 4
 Graph y ≤ 3
Horizontal line at y = 3
Use test point (0.0)  Graph x ≥ 4
Vertical line at x = 4
Use test point (1,0)
Graph both together to find area of overlap
Solve:
y ≤ [x/2] − 3
y < 5
y ≤ [x/2] − 3
y < 5
 Graph y ≤ [x/2] − 3b = − 3m = [1/2]
Use test point (0.0)  Graph y < 5
Horizontal line at y = 5
Use test point (1,0)
Graph both together to find area of overlap
Solve:
y ≤ 3x − 9
y ≥ − [x/3]
y ≤ 3x − 9
y ≥ − [x/3]
 Graph y ≤ 3x − 9b = − 9m = 3
Use test point (0.0)  Graph y ≥ − [x/3]b = 0m = − [1/3]
Use test point (1,0)
Graph both together to find area of overlap
Solve:
y ≤ − 4
y > 1
y ≤ − 4
y > 1
 Graph y ≤ − 4
Horizontal line at y = 4
Use test point (0.0)  Graph y > 1
Horizontal line at y = 1
Use test point (1,0)
Graph both together to find area of overlap
No Solutions
No Solutions
Solve:
 

 Graph y > 4x  2
b =  2
m = 4
Use test point (0.0  Graph y > x + 7
b = 7
m = 1
Use test point (1,0)
Graph both together to find area of overlap
Solve:
y + x < 7
2y + 2x >− 4
y + x < 7
2y + 2x >− 4
 Graph y + x < 7x = 0 → (0,7)y = 0 → (7,0)
Use test point (0.0)  Graph 2y + 2x >− 4x = 0 → (0, − 2)y = 0 → ( − 2,0)
Use test point (0.0)
Graph both together to find area of overlap
Solve:
2y − 3x ≤ − 6
6y + 4x > 12
2y − 3x ≤ − 6
6y + 4x > 12
 Graph 2y − 3x ≤ − 6x = 0 → (0, − 3)y = 0 → (2,0)
Use test point (0.0)  Graph 6y + 4x > 12x = 0(0,2)y = 0(3,0)
Use test point (0.0)
Graph both together to find area of overlap
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Graphing Systems of Inequalities
Lecture Slides are screencaptured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
 Intro 0:00
 System of Inequalities 0:05
 Example
 Solving a System of Inequalities 0:38
 Solution Set
 Graph Each Inequality
 Area of Overlap
 Example 1: Solve the System of Inequalities 2:44
 Example 2: Solve the System of Inequalities 6:33
 Example 3: Solve the System of Inequalities 11:40
 Example 4: Solve the System of Inequalities 17:36
2 answers
Last reply by: Juan Herrera
Sat Aug 3, 2013 12:28 PM
Post by Juan Herrera on August 2, 2013
In the examples, how do you write or express the area that overlap solution found of the inequalities as algebraic expression?