2.1Introduction to Inequalities
Equations say two things are equal. Inequalities say one thing is greater than or less than another — and their solutions are entire sets of numbers, not just one value.
Why This Matters
Inequalities describe ranges and constraints — speed limits, budget caps, safe temperature ranges. You'll use them in Algebra 2, Precalculus, and real-world decision-making wherever a single exact answer isn't the goal.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How is the solution of an inequality different from the solution of an equation, and how do we represent that solution on a number line?
Lesson Overview
An inequality is a mathematical statement that compares two expressions using one of four symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike an equation, which has at most one solution, an inequality typically has infinitely many solutions — an entire range of values that make the statement true. We represent this solution set by graphing on a number line: an open circle () means the endpoint is not included (strict inequality), while a closed circle () means the endpoint is included. The shaded ray shows all values in the solution set.
Quick Reference — All Four Inequality Symbols
| Symbol | Meaning | Circle | Shade | Example |
|---|---|---|---|---|
| < | Less than | ○ Open | ← Left | x < 3 |
| > | Greater than | ○ Open | → Right | x > 3 |
| ≤ | Less than or equal to | ● Closed | ← Left | x ≤ 3 |
| ≥ | Greater than or equal | ● Closed | → Right | x ≥ 3 |
Worked Examples
Write the inequality shown by the graph: open circle at 4, shaded to the right.
Step 1 — Read the circle: Open circle → endpoint NOT included → use strict inequality (< or >).
Step 2 — Read the direction: Shaded to the RIGHT → values greater than the endpoint.
Step 3 — Write the inequality: x > 4
Step 4 — Interval notation: (4, ∞)
Check: Test x = 6: 6 > 4 ✓. Test x = 2: 2 > 4 ✗ (not in solution set — correct).
Graph: x > 4
Open circle at 4 (not included) · shaded right → (4, ∞)
Graph x ≤ −2 on a number line and write in interval notation.
Step 1 — Identify the symbol: ≤ means "less than or equal to" → closed circle (endpoint included).
Step 2 — Mark the endpoint: Draw a closed circle (●) at −2.
Step 3 — Shade the direction: ≤ means less than → shade LEFT (toward −∞).
Step 4 — Interval notation: (−∞, −2] — the bracket ] means −2 is included.
Check: Test x = −5: −5 ≤ −2 ✓. Test x = 0: 0 ≤ −2 ✗ (not shaded — correct).
Graph: x ≤ −2
Closed circle at −2 (included) · shaded left → (−∞, −2]
A roller coaster requires riders to be at least 48 inches tall. Write an inequality for the height h of a rider who may ride.
Step 1 — Identify the key phrase: "At least 48 inches" → the height must be 48 or more.
Step 2 — "At least" means ≥: h ≥ 48
Step 3 — Graph: Closed circle at 48, shade right.
Step 4 — Interpret: Any height of 48 inches or taller is in the solution set.
Check: 52 ≥ 48 ✓ (may ride). 45 ≥ 48 ✗ (cannot ride).
Graph: h ≥ 48 (height in inches)
Closed circle at 48 · shaded right → [48, ∞)
Is x = 3 a solution of x < 3? Is x = 3 a solution of x ≤ 3?
Test x = 3 in x < 3: 3 < 3 → FALSE. So x = 3 is NOT a solution of x < 3.
Test x = 3 in x ≤ 3: 3 ≤ 3 → TRUE. So x = 3 IS a solution of x ≤ 3.
Key insight: The "or equal to" part is what makes the difference. Open circle excludes; closed circle includes.
x < 3 — open circle, x = 3 excluded
3 < 3 is FALSE → x = 3 is NOT a solution
x ≤ 3 — closed circle, x = 3 included
3 ≤ 3 is TRUE → x = 3 IS a solution
Marcus can spend no more than $35 on lunch for the week. Write an inequality for the amount a he can spend.
Step 1 — Identify the key phrase: "No more than $35" → the amount must be 35 or less.
Step 2 — "No more than" means ≤: a ≤ 35
Step 3 — Graph: Closed circle at 35, shade left.
Step 4 — Interpret: Marcus can spend $0 up to and including $35.
Check: $30 ≤ 35 ✓. $40 ≤ 35 ✗ — over budget.
Graph: a ≤ 35 (dollars)
Closed circle at 35 · shaded left → (−∞, 35]
Guided Practice
Guided Practice Video: Introduction to Inequalities
Watch the guided practice walkthrough for introduction to inequalities, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Write the inequality for: open circle at 7, shaded to the left.
Hint: Open circle → strict inequality (< or >). Shaded left → 'less than'. The endpoint is 7.
Given graph — write the inequality:
Graph x ≥ −3 on a number line and write in interval notation.
Hint: ≥ includes the endpoint → closed circle (●). ≥ means greater than → shade right. Mark ● at −3, draw arrow pointing right.
Draw your graph here:
Is x = 5 a solution of x > 5? Justify with substitution.
Hint: Substitute x = 5 into x > 5: 5 > 5 is FALSE. Strict inequality: 5 is NOT greater than itself.
Translate: 'A student needs more than 70 points to pass.' Write an inequality for points p.
Hint: "More than" → strict inequality (>). The variable is p (points).
Graph your answer:
Write the inequality shown: closed circle at 0, shaded to the right.
Hint: Closed circle → includes endpoint → ≤ or ≥. Shaded right → greater than or equal to. Endpoint is 0.
Given graph — write the inequality:
Key Vocabulary
Inequality
A mathematical statement comparing two expressions using <, >, ≤, or ≥.
Example: x > 3 means x can be any number strictly greater than 3.
Solution Set
All values of the variable that make the inequality true — usually an infinite set.
Example: The solution set of x < 5 includes 4, 3, 0, −2, and infinitely many others.
Open Circle ○
Endpoint NOT included — used for strict inequalities (< or >).
x > 4: open circle at 4
Closed Circle ●
Endpoint IS included — used for ≤ or ≥.
x ≥ 4: closed circle at 4
Interval Notation
A compact way to write a solution set using parentheses (excluded) and brackets (included).
Example: x > 3 → (3, ∞); x ≤ 7 → (−∞, 7]
At Least / At Most
"At least" means ≥ (includes the value); "at most" means ≤ (includes the value).
Example: "At least 18" → x ≥ 18. "At most 100" → x ≤ 100.
Interval Notation — Visual Reference
Interactive Practice — 5 Questions
Which graph correctly represents x > 3?
What type of circle is used to graph x ≥ −2?
Which inequality matches the phrase "at most 20"?
Is x = 5 a solution of x < 5?
Which interval notation represents x ≤ 7?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Write the inequality: open circle at −5, shaded right.
Write the inequality: closed circle at 2, shaded left.
Graph x > 6 on a number line. Write in interval notation.
Translate: "A car must travel at most 65 mph on the highway." Write an inequality for speed s.
Is x = −3 a solution of x ≥ −3? Justify with substitution.
Problem 1 — graph space:
Problem 2 — graph space:
Problem 3 — graph space:
Common Mistakes
Using a closed circle (●) for strict inequalities like x > 3 or x < 5.
Strict inequalities (< or >) use an open circle (○). Use a closed circle (●) only for ≤ or ≥.
Shading the wrong direction — e.g., shading left for x > 3.
For x > 3, shade to the right (greater values). For x < 3, shade to the left.
Writing interval notation with the wrong bracket — e.g., [3, ∞) for x > 3.
x > 3 uses a parenthesis at 3: (3, ∞). Use brackets only when the endpoint is included (≤ or ≥).
Confusing "at least" with "more than" — treating them as the same symbol.
"At least 5" means ≥ 5 (includes 5); "more than 5" means > 5 (excludes 5). The "or equal to" is the difference.
Visual: Open vs. Closed Circle
Open Circle ○
Endpoint NOT included
Use for < or >
x > 3
Closed Circle ●
Endpoint IS included
Use for ≤ or ≥
x ≥ 3
Visual: Correct vs. Incorrect Shading Direction for x > 3
✗ Incorrect — wrong direction
x > 3
Shaded left — but greater than means shade right.
✓ Correct
x > 3
Shaded right — correct for x > 3.
Math Tips
"Less than" → shade LEFT. The symbol < points left, so shade left. "Greater than" → shade RIGHT. The symbol > points right.
"Or equal to" → closed circle (●). No "equal to" → open circle (○). The "or equal to" in ≤ and ≥ is what includes the endpoint.
Always check your answer: pick any number in your shaded region and substitute it into the original inequality. It should make a true statement.
SAT/ACT tip: "at least" means ≥; "at most" means ≤; "more than" means >; "fewer/less than" means <. Memorize these four phrases.