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AlgebraCore

Linear Inequalities in One or Two Variables

Handle inequalities like equations — with one critical exception — and read shaded-region graphs with confidence.

⏱ 7 min📍 Learn first: Systems of Two Linear Equations
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Learn Linear Inequalities in One or Two Variables in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:32

Linear Inequality Signs and Absolute Values

A short spoken refresher on this concept.

00:00 / 22:56

Life inside the math of inequalities

The full lesson, worked through in depth.

Concept Snapshot

Definition
Statements like 3x + 2 ≤ 11 or y > 2x − 1. Solutions are ranges (one variable) or half-planes (two variables).
What the SAT tests
Solving and interpreting inequalities, matching systems of inequalities to shaded graphs, and word-problem constraints.
Key rule
Solve like an equation — but FLIP the inequality sign whenever you multiply or divide by a negative.
Most common mistake
Forgetting the flip, or shading above a line when the inequality says below.
Prerequisite
Systems of Two Linear Equations

Visual Explanation

What to picture: Solve like an equation — but FLIP the inequality sign whenever you multiply or divide by a negative.

A diagram for this concept is in production. Until it is ready, the worked example below shows the same relationship step by step.

Concept Mind Map

DefinitionStatements like 3x + 2 ≤ 11 or y > 2x − 1. Solutions are ranges (one variable) or half-planes (two variables).
Key ruleSolve like an equation — but FLIP the inequality sign whenever you multiply or divide by a negative.

Linear Inequalities in One or Two Variables

Algebra

Solve and FlipIsolate the variable; negatives reverse the sign.
Graph MatchingMatch a system of inequalities to its shaded region.
Constraint StoriesBudget or capacity limits become at most / at least inequalities.

How the SAT Tests It

Pattern

Solve and Flip

Isolate the variable; negatives reverse the sign.

Clue: Negative coefficient on the variable
Pattern

Graph Matching

Match a system of inequalities to its shaded region.

Clue: "which graph represents"
Pattern

Constraint Stories

Budget or capacity limits become at most / at least inequalities.

Clue: "at most", "at least", "no more than"

Worked Example

Solve: 14 − 3x ≥ 5. What is the solution set?

Isolate x, watching for the negative-division flip.
Subtract 14: −3x ≥ −9.
Divide by −3 — a negative — so the sign reverses: x ≤ 3.
x = 0: 14 − 0 = 14 ≥ 5 ✓ (0 is in x ≤ 3). x = 5: 14 − 15 = −1, not ≥ 5 ✓ excluded.

Try It Yourself

A theater has 250 seats. Adult tickets cost $12 and student tickets cost $7, and the theater needs at least $2,100 in revenue. If a adults and s students attend, which system represents the constraints?

Common Traps

What students do

Translating "at least" as ≤.

Why it fails

"At least" feels like a limit, but it's a floor — the value can be that number OR MORE.

Do this instead

Anchor the phrases: at least → ≥, at most → ≤. Test with a number: at least 5 includes 6? Yes → ≥.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Life inside the math of inequalities

Sign flips, shading rules, and constraint translation.

00:00 / 22:56

22:56 · the full lesson

What to notice

Solve like an equation — but FLIP the inequality sign whenever you multiply or divide by a negative.

What the lesson covers

  • 0:00 The flip rule
  • 1:30 At least vs at most
  • 2:50 Shaded regions
  • 4:20 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Linear Inequalities in One or Two Variables notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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