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Introduction
Inequalities with Variables on Both Sides helps you connect a mathematical rule to a result you can explain and check. Gather variable terms on one side and constants on the other. Check for sign reversal at the final division. 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 inequalities with variables on both sides.
- 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
Solve equations and compare signed numbers.
Previous lesson: Solving Multi-Step Inequalities
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- constant
- A value fixed within a problem.
- solution
- A value making the original conditions true.
- 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.
Inequalities describe solution sets
Gather variable terms on one side and constants on the other. Check for sign reversal at the final division.
An inequality compares quantities using <, >, ≤, or ≥. Its solution is usually a set of numbers. Adding or subtracting the same quantity on both sides preserves the comparison. Multiplying or dividing both sides by a negative number reverses the inequality.
The reversal follows from order on the number line: 2<5, but −2>−5. Isolate the variable, then represent all solutions. Use an open endpoint for a strict comparison and a closed endpoint when equality is included. For a context involving counts, restrict to whole numbers and respect nonnegative quantities.
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. Distribute: 12x+312≤13x+316.
- 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. Both sides agree at the endpoint; increasing x makes the right side grow faster.
- 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
Solve 12(x + 26) ≤ 13x + 316.
- Distribute: 12x+312≤13x+316.
- Subtract 12x and then 312: 0≤x+4.
- Subtract 4: −4≤x, or x≥−4. A closed endpoint is included.
Answer: x ≥ -4
Verify: Both sides agree at the endpoint; increasing x makes the right side grow faster.
Avoid this mistake: Forgetting to reverse after division by a negative number.
Example 2 · Basic
Solve 13(x + 26) ≤ 14x + 342.
- Distribute: 13x+338≤14x+342.
- Subtract 13x and then 338: 0≤x+4.
- Subtract 4: −4≤x, or x≥−4. A closed endpoint is included.
Answer: x ≥ -4
Verify: Both sides agree at the endpoint; increasing x makes the right side grow faster.
Avoid this mistake: Using an open endpoint for ≤ or ≥.
Example 3 · Intermediate
Solve 14(x + 26) ≤ 15x + 368.
- Distribute: 14x+364≤15x+368.
- Subtract 14x and then 364: 0≤x+4.
- Subtract 4: −4≤x, or x≥−4. A closed endpoint is included.
Answer: x ≥ -4
Verify: Both sides agree at the endpoint; increasing x makes the right side grow faster.
Avoid this mistake: Reporting a boundary as the only solution rather than a whole set.
Example 4 · Advanced
Solve 2(x + 27) ≤ 3x + 58.
- Distribute: 2x+54≤3x+58.
- Subtract 2x and then 54: 0≤x+4.
- Subtract 4: −4≤x, or x≥−4. A closed endpoint is included.
Answer: x ≥ -4
Verify: Both sides agree at the endpoint; increasing x makes the right side grow faster.
Avoid this mistake: Forgetting to reverse after division by a negative number.
Example 5 · Challenge
Solve 3(x + 27) ≤ 4x + 85.
- Distribute: 3x+81≤4x+85.
- Subtract 3x and then 81: 0≤x+4.
- Subtract 4: −4≤x, or x≥−4. A closed endpoint is included.
Answer: x ≥ -4
Verify: Both sides agree at the endpoint; increasing x makes the right side grow faster.
Avoid this mistake: Using an open endpoint for ≤ or ≥.
Common mistakes and corrections
- Forgetting to reverse after division by a negative number.
- Using an open endpoint for ≤ or ≥.
- Reporting a boundary as the only solution rather than a whole set.
Real-world applications
- Determine affordable ticket counts under a budget. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Describe measurements that meet a minimum safety requirement. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Gather variable terms on one side and constants on the other. Check for sign reversal at the final division.
An inequality compares quantities using <, >, ≤, or ≥. Its solution is usually a set of numbers. Adding or subtracting the same quantity on both sides preserves the comparison. Multiplying or dividing both sides by a negative number reverses the inequality.
Frequently asked questions
What should I know before inequalities with variables on both sides?
Solve equations and compare signed numbers.
How can I check my work?
Both sides agree at the endpoint; increasing x makes the right side grow faster. 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?
Forgetting to reverse after division by a negative number. Using an open endpoint for ≤ or ≥. Reporting a boundary as the only solution rather than a whole set.
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: Determine affordable ticket counts 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
- Gather variable terms on one side and constants on the other. Check for sign reversal at the final division.
- Forgetting to reverse after division by a negative number.
- Using an open endpoint for ≤ or ≥.
- Reporting a boundary as the only solution rather than a whole set.
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