The elevator says 'maximum 900kg' — may you board at exactly 900?
An inequality is the language of ranges — the answer isn't one number but a whole interval of them.
Experiment
Hands-on experiment
🔮 Predict first — the elevator says 'maximum 900kg'. May you board at exactly 900?
🛗 Change the weight and find the boundary
Load and unload cargo; find where allowed (⭕) turns into refused (❌).
700kg
⭕ OK to board (at most 900kg)
📖 Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Half the world's rules aren't equalities — 'max 900kg', 'ages 19+', 'within 30 minutes'. We needed a language for allowed ranges, not exact values.
Hence four symbols (<, >, ≤, ≥) and their words (less than, greater than, at most, at least). Whether the boundary counts (≤) or not (<) decides pass or fail in real life.
An inequality's solution is an interval on the number line. Computing 'how far is allowed' — budgeting, comparing plans, safety limits — is solving inequalities.
Insight
Insights from the video
“An inequality's answer is a region, not a point.”
An equation finds the scale's single balance point; an inequality finds the entire allowed side. The moment you shade the line, thinking expands from one number to a range.
“Multiplying by a negative is a mirror.”
×(−1) flips the number line about 0 — what was bigger becomes smaller. See 3 > 2 flip into −3 < −2 as a picture, and the sign-flip stops being memorization.
Misconception
Common misconceptions
The answer to x < 3 is 2.
Not just 2 — also 1, 0, 2.9, −100… every number below 3. An inequality's solution is a range: an entire ray on the number line. Getting used to 'infinitely many answers' is step one.
Like equations, you may multiply both sides by anything.
Multiplying by a negative flips the sign: 3 > 2, yet −3 < −2 — flipping the line about 0 swaps left and right. This single exception is all of inequality arithmetic.
Formula
Writing it as math
What the elevator experiment confirmed, in mathematical language.
The four symbols
≤ includes the boundary; < excludes it. 'At most 900kg' lets 900 board; 'less than' does not.
The rules
Add or subtract anything, multiply or divide by positives — the direction holds, just like equations.
The one exception
Multiply (or divide) by a negative and the sign flips. 3 > 2 yet −3 < −2 — the logic of flipping the line.
In Real Life
Where you meet it in real life
Safety limits
Elevator capacity, ride height limits ('120cm and up'), bridge load ratings — safety speaks in inequalities, and boundary inclusion is the legal crux.
Comparing plans
'Unlimited wins if you call more than x minutes a month' — set the two cost formulas as an inequality and the switch-over line falls out.
Budgets
Buy $1.20 items with at most $10: 1.2x ≤ 10 → x ≤ 8.3 → at most 8. The rounding lab's floor meets the inequality.
Code branches
'Show a warning when HP is under 30%' — code's if (hp < 0.3) IS an inequality. Every branch in a program runs on inequality checks.
Try Yourself
Test yourself
Q1Does 'ages 19 and up' include 19? Is it the same rule as 'under 19 not admitted'?Show answer ▾
'And up' (≥) includes 19. And 'under 19 refused' (x<19 refused) equals '19 and up admitted' (x≥19) — the same rule in two phrasings; the opposite of 'under' is 'and up'.
Q2Solve 2x + 3 < 11.Show answer ▾
Subtract 3: 2x < 8; divide by 2 (positive): x < 4. The answer is every number below 4 — the whole line left of 4 (open circle at 4).
Q3What's the catch when solving −3x ≤ 9?Show answer ▾
Dividing by −3 flips the sign: x ≥ −3. Check with x=0: −3×0 = 0 ≤ 9 ✓ — and 0 is indeed ≥ −3.
💡 Try answering yourself before revealing it — getting it wrong is where learning starts.
Watch
Related video
Connection
Concepts connect
Previous concept
Equations
Handle the equal sign's balance first, and the tilt of the inequality shows.
← Equations labLeads to next
Simultaneous Equations
One condition narrowed a range; now let two conditions squeeze the answer to a single point.
Go to the Simultaneous Equations lab →Related
Labs worth exploring together
Related lab
Equations
Seek one balance point (=) and then a whole allowed range (≤) — that's the step to inequalities.
Go to the Equations lab →Related lab
Negative Numbers
Why the sign flips — meet ×(−1) as the number-line mirror first.
Go to the Negative Numbers lab →