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Introduction
Writing Linear Inequalities from Tables, Graphs, and Words helps you connect a mathematical rule to a result you can explain and check. Reconstruct a linear boundary from a table or graph, then use shading or a verbal constraint to select the inequality direction and equality inclusion. This lesson belongs to coordinate geometry and graphing 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 writing linear inequalities from tables, graphs, and words.
- 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: Transformations of Linear Functions
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
Reconstruct a linear boundary from a table or graph, then use shading or a verbal constraint to select the inequality direction and equality inclusion.
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. The boundary changes by 12 per unit input and has y-intercept 23, so y=12x + 23.
- 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 the boundary is y=b; points with y≤b satisfy the constraint.
- 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
A graph has a solid boundary through (0,23) and (1,35) and shading below the line. Write its inequality.
- The boundary changes by 12 per unit input and has y-intercept 23, so y=12x + 23.
- The allowed region is below the boundary, giving y less than its boundary output.
- Equality is included, so y≤12x + 23. A graph uses a solid line; (0,0) is a test point because 23>0.
Answer: y ≤ 12x + 23
Verify: At x=0 the boundary is y=b; points with y≤b satisfy the constraint.
Avoid this mistake: Choosing a solid boundary for a strict inequality.
Example 2 · Basic
A boundary table contains (0,23), (1,36), and (2,49). Points on or below the boundary are allowed. Write its inequality.
- The boundary changes by 13 per unit input and has y-intercept 23, so y=13x + 23.
- The allowed region is below the boundary, giving y less than its boundary output.
- Equality is included, so y≤13x + 23. A graph uses a solid line; (0,0) is a test point because 23>0.
Answer: y ≤ 13x + 23
Verify: At x=0 the boundary is y=b; points with y≤b satisfy the constraint.
Avoid this mistake: Testing a point on the boundary, which does not distinguish sides.
Example 3 · Intermediate
A plan allows total output y to be at most a starting allowance 23 plus 14 per input unit x. Write the constraint.
- The boundary changes by 14 per unit input and has y-intercept 23, so y=14x + 23.
- The allowed region is below the boundary, giving y less than its boundary output.
- Equality is included, so y≤14x + 23. A graph uses a solid line; (0,0) is a test point because 23>0.
Answer: y ≤ 14x + 23
Verify: At x=0 the boundary is y=b; points with y≤b satisfy the constraint.
Avoid this mistake: Shading according to a rearranged rule with an incorrect sign reversal.
Example 4 · Advanced
A graph has a solid boundary through (0,24) and (1,26) and shading below the line. Write its inequality.
- The boundary changes by 2 per unit input and has y-intercept 24, so y=2x + 24.
- The allowed region is below the boundary, giving y less than its boundary output.
- Equality is included, so y≤2x + 24. A graph uses a solid line; (0,0) is a test point because 24>0.
Answer: y ≤ 2x + 24
Verify: At x=0 the boundary is y=b; points with y≤b satisfy the constraint.
Avoid this mistake: Choosing a solid boundary for a strict inequality.
Example 5 · Challenge
A boundary table contains (0,24), (1,27), and (2,30). Points on or below the boundary are allowed. Write its inequality.
- The boundary changes by 3 per unit input and has y-intercept 24, so y=3x + 24.
- The allowed region is below the boundary, giving y less than its boundary output.
- Equality is included, so y≤3x + 24. A graph uses a solid line; (0,0) is a test point because 24>0.
Answer: y ≤ 3x + 24
Verify: At x=0 the boundary is y=b; points with y≤b satisfy the constraint.
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
Reconstruct a linear boundary from a table or graph, then use shading or a verbal constraint to select the inequality direction and equality inclusion.
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 writing linear inequalities from tables, graphs, and words?
Graph linear equations and solve inequalities.
How can I check my work?
At x=0 the boundary is y=b; points with y≤b satisfy the constraint. 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
- Reconstruct a linear boundary from a table or graph, then use shading or a verbal constraint to select the inequality direction and equality inclusion.
- 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