Systems of Linear Inequalities
A system of linear inequalities is a set of two or more inequalities. The solution to the system is the region where the shaded areas of all inequalities overlap.
Visualizing the Overlap
In the graph below, we are looking at two inequalities:
1. y < -x + 5 (Red region)
2. y > x – 1 (Blue region)
The solution set is the dark purple area where both colors meet!
How to Solve Systems
- Graph the first inequality: Use a solid line for ≤ or ≥, and a dashed line for < or >. Shade the appropriate side.
- Graph the second inequality: On the same coordinate plane, graph and shade the second inequality.
- Identify the Overlap: The final solution is only the area that contains shading from both inequalities.
Example: Is (0,0) a Solution?
To check if a point is a solution, plug the coordinates into both inequalities. If it makes both true, it is in the overlap!
System: y > x + 1 and y < 5
Check (0,0):
0 > 0 + 1 → False (0 is not greater than 1)
Since it failed the first one, (0,0) is NOT a solution to the system.
Mastery Quiz
1. If a system has the inequalities y ≥ 2x and y < 5, what kind of line should you draw for y < 5?
Show Answer
Answer: A dashed line. (Because the symbol is “less than” without the equal sign.)
2. Is the point (1, 2) a solution to the system: y > x and y < 4?
Show Answer
Answer: Yes. (2 > 1 is True, and 2 < 4 is True. Both are satisfied!)
3. Where is the solution to a system of inequalities located on a graph?
Show Answer
Answer: The overlapping shaded region.
Achievement Unlocked
Inequality Investigator
“You have mastered the art of the overlap and found the hidden solutions.”