Systems of Linear Inequalities
A free Algebra I lesson from the “Systems of Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.
A system of linear inequalities is solved by graphing each inequality and finding the overlap of the shaded regions. A point is a solution only when it satisfies EVERY inequality in the system — substitute and check all of them. In Systems of Linear Equations and Inequalities, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.
What you'll learn
- Graph two or more linear inequalities on the same coordinate plane
- Identify the overlapping shaded region as the solution set
- Verify whether a point satisfies an entire system
Worked example
Problem. Is (3, 1) a solution of the system y > x - 2 AND y < 2x?
- Check first: is 1 > 3 - 2 = 1? No (strictly greater fails).
- Because one inequality fails, (3, 1) is NOT a solution of the system.
- Connect the result back to Systems of Linear Inequalities so the method and meaning are both clear.
- Check the result against the original representation before writing the final answer.
Answer: No
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. Is (2, 3) a solution of {y < 5 AND y > 0}?
Choices: Yes · No
Show solution
- Check: 3 < 5 yes; 3 > 0 yes.
- Both hold, so (2, 3) is a solution.
Answer: Yes
2. Is (3, 1) a solution of {y > x - 2 AND y < 2x}?
Choices: Yes · No
Show solution
- Check first: 1 > 3 - 2 = 1 fails.
- Strict > is not satisfied, so not a solution.
Answer: No
3. Is (1, 2) a solution of {y > x AND y < 2x + 1}?
Choices: Yes · No
Show solution
- Check: 2 > 1 yes; 2 < 3 yes.
- Both hold.
Answer: Yes
4. Is (0, 0) a solution of {y >= 0 AND x >= 0}?
Choices: Yes · No
Show solution
- Both 0 >= 0 statements hold.
- So (0, 0) is on the boundary of both and counts.
Answer: Yes
5. Is (-1, 2) a solution of {y > 0 AND x > 0}?
Choices: Yes · No
Show solution
- First: 2 > 0 yes. Second: -1 > 0 NO.
- Fails the x condition.
Answer: No
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