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Introduction
Graphing Linear Inequalities in Two Variables helps you connect a mathematical rule to a result you can explain and check. Draw the boundary with the correct style, test a point away from it, and shade the half-plane that satisfies the original inequality. 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 graphing linear inequalities in two variables.
- 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 linear equations and solve inequalities.
Previous lesson: Writing Inequalities from Word Problems
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- coefficient
- A factor multiplying a variable term.
- equation
- A statement that two expressions are equal.
- inequality
- A comparison using <, >, ≤, or ≥.
- 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.
Graph a half-plane
Draw the boundary with the correct style, test a point away from it, and shade the half-plane that satisfies the original inequality.
A two-variable inequality usually describes a half-plane containing infinitely many ordered pairs. First graph its boundary equation. Use a solid line for ≤ or ≥ and a dashed line for < or >.
Choose a test point away from the boundary, substitute it into the original inequality, and shade its side only when it makes the comparison true. If you divide by a negative coefficient while isolating y, reverse the inequality. Points on a dashed boundary are excluded even when close to the shaded area.
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. Replace ≤ by = to draw the boundary line with slope 12 and intercept 29.
- 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. At x=0, boundary height is b>0; (0,0) lies in the shaded half-plane.
- 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
Graph y ≤ 12x + 29. Which boundary and side are correct?
- Replace ≤ by = to draw the boundary line with slope 12 and intercept 29.
- Equality is included, so use a solid line.
- Test (0,0): 0≤29 is true. Shade the side below the line, including the boundary.
Answer: Solid boundary y = 12x + 29; shade below
Verify: At x=0, boundary height is b>0; (0,0) lies in the shaded half-plane.
Avoid this mistake: Choosing a solid boundary for a strict inequality.
Example 2 · Basic
Graph y ≤ 13x + 29. Which boundary and side are correct?
- Replace ≤ by = to draw the boundary line with slope 13 and intercept 29.
- Equality is included, so use a solid line.
- Test (0,0): 0≤29 is true. Shade the side below the line, including the boundary.
Answer: Solid boundary y = 13x + 29; shade below
Verify: At x=0, boundary height is b>0; (0,0) lies in the shaded half-plane.
Avoid this mistake: Testing a point on the boundary, which does not distinguish sides.
Example 3 · Intermediate
Graph y ≤ 14x + 29. Which boundary and side are correct?
- Replace ≤ by = to draw the boundary line with slope 14 and intercept 29.
- Equality is included, so use a solid line.
- Test (0,0): 0≤29 is true. Shade the side below the line, including the boundary.
Answer: Solid boundary y = 14x + 29; shade below
Verify: At x=0, boundary height is b>0; (0,0) lies in the shaded half-plane.
Avoid this mistake: Shading according to a rearranged rule with an incorrect sign reversal.
Example 4 · Advanced
Graph y ≤ 2x + 30. Which boundary and side are correct?
- Replace ≤ by = to draw the boundary line with slope 2 and intercept 30.
- Equality is included, so use a solid line.
- Test (0,0): 0≤30 is true. Shade the side below the line, including the boundary.
Answer: Solid boundary y = 2x + 30; shade below
Verify: At x=0, boundary height is b>0; (0,0) lies in the shaded half-plane.
Avoid this mistake: Choosing a solid boundary for a strict inequality.
Example 5 · Challenge
Graph y ≤ 3x + 30. Which boundary and side are correct?
- Replace ≤ by = to draw the boundary line with slope 3 and intercept 30.
- Equality is included, so use a solid line.
- Test (0,0): 0≤30 is true. Shade the side below the line, including the boundary.
Answer: Solid boundary y = 3x + 30; shade below
Verify: At x=0, boundary height is b>0; (0,0) lies in the shaded half-plane.
Avoid this mistake: Testing a point on the boundary, which does not distinguish sides.
Common mistakes and corrections
- Choosing a solid boundary for a strict inequality.
- Testing a point on the boundary, which does not distinguish sides.
- Shading according to a rearranged rule with an incorrect sign reversal.
Real-world applications
- Show combinations of two products affordable under a budget. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Describe combinations of hours meeting a minimum earnings goal. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Draw the boundary with the correct style, test a point away from it, and shade the half-plane that satisfies the original inequality.
A two-variable inequality usually describes a half-plane containing infinitely many ordered pairs. First graph its boundary equation. Use a solid line for ≤ or ≥ and a dashed line for < or >.
Frequently asked questions
What should I know before graphing linear inequalities in two variables?
Graph linear equations and solve inequalities.
How can I check my work?
At x=0, boundary height is b>0; (0,0) lies in the shaded half-plane. 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?
Choosing a solid boundary for a strict inequality. Testing a point on the boundary, which does not distinguish sides. Shading according to a rearranged rule with an incorrect sign reversal.
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: Show combinations of two products affordable under a budget.
- 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
- Draw the boundary with the correct style, test a point away from it, and shade the half-plane that satisfies the original inequality.
- Choosing a solid boundary for a strict inequality.
- Testing a point on the boundary, which does not distinguish sides.
- Shading according to a rearranged rule with an incorrect sign reversal.
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