An equation asks a narrow question and hands back one number. An inequality asks a wider one — which numbers satisfy this condition? — and the answer is usually an infinite shaded stretch of the number line.
⚖️Four Symbols, One Rule
The wide open end always faces the larger value; the sharp point aims at the smaller one. Add a bar underneath and the two sides are also allowed to match.
📏Graph the Solution Set
The answer is a set, not a number. Mark the boundary (found by treating the inequality as an equation), choose your dot, then shade the direction that works — and always test one value from the shaded side.
🧭The One Move That Reverses the Symbol
Every legal move you trust on equations works here too — with exactly one exception. Multiplying or dividing both sides by a negative number flips the symbol, because negation reflects the entire number line across zero, and left-to-right order becomes right-to-left.
🔄 FLIP · −3x ≥ 18
−3x ≥ 18the coefficient is negative
x ≤ −6divide by −3 — and flip ≥ into ≤
Test it. x = −10 → −3(−10) = 30 ≥ 18 ✅. x = 0 → 0 ≥ 18 ❌. The shading really does go left.
✅ NO FLIP · 4x − 9 > 11
4x > 20add 9 to both sides — adding never flips
x > 5divide by positive 4 — symbol unchanged
Test it. x = 6 → 24 − 9 = 15 > 11 ✅. x = 5 → 11 > 11 ❌ (which is why the circle is open).
🧮One more with a hidden negative: 5 − 2x ≤ 13. Subtract 5 → −2x ≤ 8. Divide by −2 and flip → x ≥ −4. Check: x = 0 gives 5 ≤ 13 ✅, but x = −10 gives 5 + 20 = 25 ≤ 13 ❌. The flip was real.
⚠️Adding or subtracting a negative NEVER flips the symbol. Only multiplying or dividing by a negative does. And you may never multiply or divide an inequality by a variable whose sign you don’t know — you wouldn’t know whether to flip.
🔗Compound Inequalities: AND vs OR
AND requires both conditions at once, so its solution set is the overlap. OR requires only one, so its solution set is the union of two regions.
💭Words → Symbols → Intervals
Constraints are the reason inequalities matter outside a textbook. In interval notation, a square bracket includes the endpoint, a parenthesis excludes it — and infinity always takes a parenthesis, because you never arrive there.
The real limit
Inequality
Interval
A backpack carries at most 15 lb
w ≤ 15
(−∞, 15]
A ride requires height at least 48 in.
h ≥ 48
[48, ∞)
Fewer than 20 people in the room
p < 20
(−∞, 20)
More than $60 raised
m > 60
(60, ∞)
Bolt spec 10.00 mm ± 0.05 mm
9.95 ≤ d ≤ 10.05
[9.95, 10.05]
The AND example above
−5 < x ≤ 2
(−5, 2]
🔧A tolerance is a compound inequality in disguise. An inspector’s whole job is testing whether a measured diameter lands inside that interval. Anything past the boundary is scrap — no matter how close it came. And a household budget is not an equation either: you almost never spend your income to the penny. It’s total spending ≤ money available — which is exactly why the phrase “debt-to-income ratio” is followed by a ≤ and a threshold number.
🔑Key Terms
⚖️InequalityCompares two expressions with <, >, ≤, or ≥ instead of =. Its answer is a range of values, not a single number.
🎨Solution SetEvery value that makes the inequality true, gathered together. For a linear inequality this is an infinite interval — hence the shading.
📏Number LineEvery real number has a position, increasing left to right. It is what makes the direction of an inequality symbol meaningful.
🚩BoundaryThe edge value, found by treating the inequality as an equation. Filled dot when included (≤, ≥); open dot when not (<, >).
🔗Compound InequalityTwo inequalities in one statement. AND → the overlap (often written 18 < T < 27). OR → the union of two regions.
🔄Flip Rule (Reversal)Multiplying or dividing both sides by a negative reverses the symbol, because negation reflects the number line across zero.
📝Interval NotationShorthand for a solution set. [ ] includes the endpoint, ( ) excludes it, and ∞ always takes a parenthesis.
🔒ConstraintA real-world limit written as an inequality — a maximum budget, a minimum height, a weight capacity, a safe temperature range.
🌎Worth Knowing
📜< and > first appeared in print in 1631, in Thomas Harriot’s Artis Analyticae Praxis — a book published ten years after Harriot had died. The “or equal to” versions, ≤ and ≥, didn’t arrive for another century; they’re usually credited to Pierre Bouguer in 1734.
✈️Inequalities are the engine behind linear programming — the branch of math that finds the best possible outcome inside a wall of constraints. George Dantzig’s simplex method (1947) still schedules airline crews, routes delivery trucks, and blends animal feed at minimum cost every single day.
🔄One more operation flips the symbol: taking reciprocals — as long as both sides share the same sign. Since 2 < 5, it follows that 1/2 > 1/5. Try it with two negatives and the same flip happens.
📌Remember This
1Same legal moves as an equation — with exactly one exception: multiplying or dividing both sides by a negative reverses the symbol, because negation reflects the number line across zero.
2The answer is a solution set, not a single number. Open circle for < and >, filled circle for ≤ and ≥ — then shade the direction that works, and test a value to be sure.
3Real limits translate directly: “at most” / “no more than” → ≤, “at least” / “no fewer than” → ≥, and a limit with both a floor and a ceiling becomes a compound inequality joined by AND.
🤔 Think about it
Multiplying 2 < 5 by any positive number keeps it true, and by any negative number makes it true again once you flip. But multiply both sides by 0 and you get 0 < 0 — false, and no flip can rescue it. What is special about zero that breaks the pattern — and what does that tell you about multiplying an inequality by a variable whose sign you don’t know?
Solve x² > 9 by taking the square root of both sides and you’ll write x > 3 — but test x = −10: (−10)² = 100, which really is greater than 9. Sketch y = x² and the line y = 9 on the same axes. What does the picture reveal that the one-step algebra hid from you?
⭐Remember: an inequality’s answer is a whole shaded region, and the boundary is the exact line between allowed and turned away. Solve it like an equation — then watch for that one negative that turns the world around.
✏️ ClickClass Stretch Anchor Chart · Algebra I · Inequalities