Inequalities:
a whole range
of answers.

Four little signs, and how to read them out loud.

An inequality is just two amounts with a sign between them saying how they compare. There are four signs to know, and the trick to never mixing them up is to read the wide open end as "the big side" — the symbol always opens its mouth toward the larger number, like a hungry crocodile going for the bigger snack.

Now the everyday words, because this is where inequalities actually live. "At least" means "this much or more" — a floor you can't go under — and that's ≥. "At most" means "this much or less" — a ceiling you can't go over — and that's ≤. The little line under the arrow is doing one job: it quietly adds "...or equal to," which is what lets the boundary number itself join the club.

Watch the difference between the loose words and the strict ones. "More than 100" and "over 100" mean > 100 — 100 is not included. But "at least 100" and "100 or more" mean ≥ 100 — now 100 counts. Same with the small side: "fewer than 30" and "under 30" are < 30, while "at most 30" and "no more than 30" are ≤ 30. That one word — "at least," "at most" — decides whether the boundary is in or out, and in the next demo you'll see it as the difference between a hollow circle and a filled one.

Build the solution on a number line.

This is the heart of the page. Pick one of the four symbols, slide the boundary to any value, and watch the line shade in every number that makes the inequality true. The circle on the boundary tells you whether the boundary itself is in the club: a hollow (open) circle means "not included," a filled (closed) circle means "included." Then drag the test marker to drop any number onto the line and ask: is it in the shaded set, yes or no?

Play with it until the picture feels obvious: the symbol decides which direction the shading runs and whether the circle is open or closed; the boundary decides where the shading starts. Everything to the shaded side, all the way to the arrow, is a number that works. That arrow matters — it's reminding you the solution never stops. There's no biggest number in x > 3, the same way there's no biggest number, full stop.

Why one circle is hollow and the other is filled.

That little circle on the boundary is the most important dot on the whole line, so it's worth slowing down on. The boundary number sits right on the fence between "works" and "doesn't work," and the circle is the page's way of telling you which side of the fence the boundary itself falls on.

With < and > — the strict signs, with no line underneath — the boundary is excluded. Think about x > 3: that says x is strictly bigger than 3, and 3 is not bigger than itself, so 3 doesn't qualify. We draw a hollow circle right on 3 to say "close, but this exact point is left out." Everything just past it is shaded; 3 itself is a tiny gap.

With ≤ and ≥ — the signs wearing the extra line — the boundary is included, because the symbol literally says "or equal to." For x ≥ 3, the value 3 is allowed (it's "3 or more"), so we fill the circle in solid: 3 is part of the answer, and so is everything above it. The filled circle means "yes, start counting from here, this point included."

Here's a memory hook that never fails: the line under the symbol (the extra stroke in ≤ and ≥) becomes the fill inside the circle. No line under the symbol → no fill in the circle → hollow. There's a line → the circle fills in. Symbol and picture are telling you exactly the same thing, twice, so you can always check one against the other.

Solve an inequality just like an equation.

Here's the good news that makes the whole topic easy: solving an inequality works almost exactly like solving an equation. You still want to get x alone, and you still do it by undoing operations — doing the same thing to both sides — until x stands by itself. The only thing that changes is that the equals sign is now an inequality sign, and it comes along for the ride.

Take x + 2 > 5. To free x you subtract 2 from both sides, the same move you'd make for x + 2 = 5. The left becomes x, the right becomes 3, and the > sign stays put: x > 3. That's the answer — not a single number, but the whole range you already know how to draw. Pick an inequality below and watch it come apart, step by step, then see the solution land on a number line.

Notice these are the same moves you already use on equations: subtract to undo adding, add to undo subtracting, divide to undo multiplying. 2x < 8 becomes x < 4 when you divide both sides by 2; 3x + 1 ≥ 10 takes two steps — subtract 1, then divide by 3 — to reach x ≥ 3. Dividing by a positive number leaves the sign exactly as it was. There's just one situation where the sign does something surprising, and it's important enough to get its own section — next.

One rule that's different: the negative flip.

Almost everything about solving inequalities is identical to solving equations, with one exception worth tucking away: if you multiply or divide both sides by a negative number, the inequality sign flips around. The < becomes >, the ≥ becomes ≤. Don't worry about mastering this today — just meet it, so it isn't a shock later.

Why would the sign flip? Because multiplying by a negative reverses order on the number line — it spins the line around. Here's a true statement everyone agrees on: 2 < 5 (2 is less than 5). Now multiply both sides by −1. The left becomes −2, the right becomes −5. And on the number line, −2 is actually bigger than −5 (it's closer to zero, further right). So the true statement is now −2 > −5 — the sign had to flip to stay true. Tap below to see it.

"You must be at least…" — inequalities in the wild.

Inequalities aren't a school-only thing; they're how the world writes its rules. Almost any rule with the words "at least," "at most," "minimum," "maximum," "over," or "under" is secretly an inequality with a boundary and a shaded side. Read each one, decide on the symbol and whether the boundary is in or out, then tap to check.

See the pattern? The phrase tells you the symbol, the number is the boundary, and "at least / at most" versus "more than / fewer than" tells you whether the circle is filled or hollow. Once you can translate the words into a sign and a circle, you can picture the rule — every value that's allowed, and every one that isn't — on a single line.

The trap: "x > 3 means x = 4."

This is the single most common slip with inequalities, and it comes from a year of solving equations where the answer really was one number. So the mind reaches for the nearest tidy number — "x > 3? oh, that's 4" — and quietly throws away the entire rest of the range. But that's exactly the point an inequality is making: it is not one number.

The decimals are the part people forget, so it's worth saying plainly: there is no "first number after 3." You might think x > 3 starts at 4, but 3.5 beats it, and 3.1 beats that, and 3.01 beats that — you can always squeeze in a closer number, forever. That's why we don't list the answers to an inequality; we shade them. The number line isn't decoration — it's the only honest way to show "all of these, with no gaps," because there are far too many to ever write down.

Four quick checks.

No pressure and no streak to protect — just four questions to see if the symbols, the circles, and the solving have clicked. Pick an answer and you'll get an explanation either way.