≤ · Inequalities

The language of ranges — the inequality.

The elevator says 'maximum 900kg' — may you board at exactly 900?

The one exception where multiplying flips the whole thing around.

Why does this question matter?

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.

Pause and try it yourself

Make a prediction first. Change one input at a time and compare what happens with what you expected.

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 the idea from another angle

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.

Now read the definitions with their conditions

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.

Make one more distinction

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.

Where else can this idea help?

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.

Explain it in your own words

Try another question

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.

What question comes next?

Previous concept

x=?

Equations

Handle the equal sign's balance first, and the tilt of the inequality shows.

← Equations lab

Leads to next

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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 lab

x=?

Equations

Seek one balance point (=) and then a whole allowed range (≤) — that's the step to inequalities.

Go to the Equations lab →

Related lab

−n

Negative Numbers

Why the sign flips — meet ×(−1) as the number-line mirror first.

Go to the Negative Numbers lab →