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Introduction
Systems of Linear Inequalities helps you connect a mathematical rule to a result you can explain and check. Find the intersection of two half-planes. Verify a proposed point against both inequalities, including boundary inclusion. This lesson belongs to linear inequalities and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in systems of linear inequalities.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Graph a two-variable inequality and test points.
Previous lesson: Graphing Linear Inequalities in Two Variables
Key mathematical vocabulary
- solution
- A value making the original conditions true.
- inequality
- A comparison using <, >, ≤, or ≥.
- rate
- A ratio measured in units per unit.
- verify
- Check against the original conditions.
- expression
- A mathematical phrase representing a value.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Intersect solution regions
Find the intersection of two half-planes. Verify a proposed point against both inequalities, including boundary inclusion.
A system of inequalities requires every inequality to be true at the same time. Graph each boundary with the proper solid or dashed style and shade the corresponding half-plane. The overlap is the feasible region.
A point satisfying only one inequality is not a solution of the system. Boundaries may be included separately; a point on one included boundary still has to satisfy the other comparison. A feasible region may be bounded, unbounded, empty, or restricted further to whole-number points in a counting problem.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Draw both solid boundaries because equality is included.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. The point is b/2 above the lower line and b/2 below the upper line.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Find the overlap for y ≥ 12x and y ≤ 12x + 30. Is (4, 63) a solution?
- Draw both solid boundaries because equality is included.
- The feasible region must be above y=12x AND below y=12x+30.
- At x=4, the allowed y-values are 48≤y≤78. The proposed y=63 lies between them.
Answer: Yes; it lies in the strip between the two solid parallel boundaries
Verify: The point is b/2 above the lower line and b/2 below the upper line.
Avoid this mistake: Taking the union instead of the intersection.
Example 2 · Basic
Find the overlap for y ≥ 13x and y ≤ 13x + 30. Is (4, 67) a solution?
- Draw both solid boundaries because equality is included.
- The feasible region must be above y=13x AND below y=13x+30.
- At x=4, the allowed y-values are 52≤y≤82. The proposed y=67 lies between them.
Answer: Yes; it lies in the strip between the two solid parallel boundaries
Verify: The point is b/2 above the lower line and b/2 below the upper line.
Avoid this mistake: Assuming every crossing boundary point is included.
Example 3 · Intermediate
Find the overlap for y ≥ 14x and y ≤ 14x + 30. Is (4, 71) a solution?
- Draw both solid boundaries because equality is included.
- The feasible region must be above y=14x AND below y=14x+30.
- At x=4, the allowed y-values are 56≤y≤86. The proposed y=71 lies between them.
Answer: Yes; it lies in the strip between the two solid parallel boundaries
Verify: The point is b/2 above the lower line and b/2 below the upper line.
Avoid this mistake: Ignoring nonnegative and whole-number restrictions in context.
Example 4 · Advanced
Find the overlap for y ≥ 2x and y ≤ 2x + 31. Is (4, 23.5) a solution?
- Draw both solid boundaries because equality is included.
- The feasible region must be above y=2x AND below y=2x+31.
- At x=4, the allowed y-values are 8≤y≤39. The proposed y=23.5 lies between them.
Answer: Yes; it lies in the strip between the two solid parallel boundaries
Verify: The point is b/2 above the lower line and b/2 below the upper line.
Avoid this mistake: Taking the union instead of the intersection.
Example 5 · Challenge
Find the overlap for y ≥ 3x and y ≤ 3x + 31. Is (4, 27.5) a solution?
- Draw both solid boundaries because equality is included.
- The feasible region must be above y=3x AND below y=3x+31.
- At x=4, the allowed y-values are 12≤y≤43. The proposed y=27.5 lies between them.
Answer: Yes; it lies in the strip between the two solid parallel boundaries
Verify: The point is b/2 above the lower line and b/2 below the upper line.
Avoid this mistake: Assuming every crossing boundary point is included.
Common mistakes and corrections
- Taking the union instead of the intersection.
- Assuming every crossing boundary point is included.
- Ignoring nonnegative and whole-number restrictions in context.
Real-world applications
- Find product combinations satisfying budget and supply limits. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Plan work hours meeting both a minimum income and maximum time limit. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Find the intersection of two half-planes. Verify a proposed point against both inequalities, including boundary inclusion.
A system of inequalities requires every inequality to be true at the same time. Graph each boundary with the proper solid or dashed style and shade the corresponding half-plane. The overlap is the feasible region.
Frequently asked questions
What should I know before systems of linear inequalities?
Graph a two-variable inequality and test points.
How can I check my work?
The point is b/2 above the lower line and b/2 below the upper line. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Taking the union instead of the intersection. Assuming every crossing boundary point is included. Ignoring nonnegative and whole-number restrictions in context.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Find product combinations satisfying budget and supply limits.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Find the intersection of two half-planes. Verify a proposed point against both inequalities, including boundary inclusion.
- Taking the union instead of the intersection.
- Assuming every crossing boundary point is included.
- Ignoring nonnegative and whole-number restrictions in context.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials