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1. Learning Objectives
- Define a system of linear inequalities in two variables.
- Graph multiple inequalities on a coordinate plane.
- Identify the solution region (overlap) where all inequalities are satisfied.
2. Key Notes
The solution to a system is the intersection of the shaded regions of each individual inequality.
- Boundary Lines: Use dashed for < or > and solid for ≤ or ≥.
- Test Point: Use (0,0) to check which side to shade (if line doesn't pass through origin).
- Feasible Region: The final overlapping area containing all valid points.
Click the button below each question to reveal the explanation.
1. Where is the solution region for: x > 0 and y > 0?
Both variables must be positive. This corresponds to the First Quadrant.
2. True or False: A system can have "No Solution".
True. If the shaded regions of the inequalities never overlap, there is no solution. Ans: True
3. Which line type is used for y ≤ 2x + 1?
Inclusive symbols (≤ or ≥) require a Solid Line.
4. Solve: y > 3 and y < 5.
The region is a horizontal strip between the lines y=3 and y=5. Ans: 3 < y < 5
5. Is (0,0) a solution for x + y > 2 and x - y < 5?
Check first eq: 0+0 > 2 (False). Since it fails one, it fails the system. Ans: No
6. Describe the graph of x ≥ -2 and x ≤ 4.
A vertical region between x=-2 and x=4. Ans: Vertical Strip
7. What is the intersection of y > x and y < x?
A value cannot be both greater than and less than x at the same time. Ans: No Solution
8. If slopes are equal and signs are opposite (y > x+1, y < x-1), is there a solution?
No, the regions shade away from each other and never meet. Ans: Empty Set
9. Solve graphically: x ≥ 0, y ≥ 0, x + y ≤ 4. What shape is formed?
The boundaries are the x-axis, y-axis, and a diagonal line. Ans: Right Triangle
10. For y < x, which side do you shade?
Shade the region Below the dashed line y=x.
11. Is (2,3) in the solution set of y ≥ x?
3 ≥ 2 is True. Ans: Yes
12. Solve: x > -1, x < 1, y > -1, y < 1. Shape?
The intersection of these four boundaries forms a Square.
13. Does x + y ≤ 0 and x + y ≥ 0 have solutions?
Yes, only on the line where x + y = 0. Ans: The line itself
14. For x ≤ 5, is the shading left or right?
Values "less than" 5 are to the Left.
15. Is a dashed line part of the solution?
No, points on a dashed line do not satisfy the strict inequality. Ans: No
16. Solve: y > 0 and x = 2.
The solution is the vertical ray starting at (2,0) going upwards. Ans: Ray
17. Which quadrant is x < 0 and y < 0?
Both negative. Ans: Third Quadrant
18. Solve: y ≥ 2 and x ≤ 4. Corner point?
The two boundary lines meet at the point (4, 2). Ans: (4, 2)
19. If y > 5 and y < 2, what is the solution?
No number can be both > 5 and < 2. Ans: No Solution
20. What is the first step in solving a system graphically?
Graphing the individual boundary lines as if they were equations. Ans: Graph Boundaries