Look around and you'll notice something odd: almost none of the rules in real life name a single number. The sign at the rollercoaster doesn't say "you must be exactly 1.2 m tall" — it says you must be at least 1.2 m. The motorway sign doesn't demand you drive at precisely 70 mph — it says no more than 70. A film rated 15 admits anyone aged 15 or over; the airline takes any bag under 23 kg. Real rules are ranges, not points.

An inequality is the mathematical version of such a rule. Where an equation like a one-step equation pins x to a single value, an inequality is looser: it names a whole stretch of the number line. The ride rule is h \ge 1.2; the speed limit is s \le 70; the luggage rule is m < 23. Each one has infinitely many solutions — every height from 1.2 m up, every speed from 70 down. You could never list them all. But you can draw them all, in one stroke, on a number line. That drawing is what this page is about.

Four symbols, one picture

There are four inequality symbols, and every one of them is a rule about position on the number line:

• x > 3 — strictly greater than: every number bigger than 3, but not 3 itself;
• x \ge 3 — greater than or equal to: everything bigger than 3, and 3 too;
• x < 3 — strictly less than: everything below 3, not including it;
• x \le 3 — less than or equal to: everything below 3, with 3 allowed as well.

Notice how x \le -1 means -1 together with everything below it — the solutions may even slide deep into the negative numbers. A handy memory hook: the symbol is a hungry crocodile whose open mouth always points at the bigger side, and the little line under \le and \ge is half of an equals sign — "or equal to".

The drawing: one circle, one ray

To draw an inequality we mark the boundary value, then sweep a ray the way the solutions go. Two choices tell the whole story. First, the circle at the boundary:

• an open circle for < or > — the boundary value is not included (x > 3 never lets x actually equal 3);
• a filled circle for \le or \ge — the boundary value is included (x \le -1 allows x = -1 itself).

Second, the direction: the ray points toward the bigger numbers (to the right) for > and \ge, and toward the smaller numbers (to the left) for < and \le. The arrowhead on the end of the ray matters: it says "…and keep going forever". The solutions never run out.

Watch the animation below a few times — it draws a different random inequality on each replay, so you can test yourself: pause after the circle appears and predict which way the ray will sweep.

Worked examples: draw x > 2 and x \le -1

Let's do two full examples, slowly. For x > 2, ask the two questions in order. Is the boundary included? No — there's no "or equal to", so the circle at 2 is open. Which side lives? "Greater than" means the bigger numbers, so the ray runs to the right.

For x \le -1: the "or equal to" bar means -1 itself qualifies, so the circle is filled; "less than" sends the ray left, down through -2, -3, -10 and beyond. Step through the figure below and watch both drawings assemble:

Reading a drawing works exactly the same, backwards: circle first (open = strict, filled = inclusive), then direction (right = greater, left = less). Two glances and you can name any inequality on sight.

Trapped between two walls: -2 < x \le 3

Some rules fence a number in from both sides. "The water in the fish tank must be warmer than 22°C but no warmer than 26°C" is two conditions at once: t > 22 and t \le 26. Mathematicians write the pair as one double inequality, 22 < t \le 26 — read it from the middle outwards: "t is greater than 22 and at most 26".

On the number line a double inequality is a segment, not a ray: it has a circle at each end, and the ends can be different styles. Take -2 < x \le 3: the left end at -2 is open (strict), the right end at 3 is filled (inclusive), and the shading joins them. Step through it:

Which integers satisfy -2 < x \le 3? Walk the segment and collect the whole numbers: -1, 0, 1, 2, 3. Note the two ends carefully — 3 is in (filled circle) but -2 is out (open circle). Five integers, even though the full set of solutions (fractions, decimals and all) is infinite.

One more for practice: list the integers satisfying -3 \le x < 2. This time the left end is the inclusive one, so start at -3 itself and stop before 2: -3, -2, -1, 0, 1. Five again — always check each end against its own symbol, because exam questions love to swap them.

From words to inequalities

Most inequality questions arrive dressed up in words, so you need the phrasebook. English has many ways of saying each symbol:

• at least, no less than, a minimum of → \ge (filled circle, ray right);
• more than, over, above, exceeds → > (open circle, ray right);
• at most, no more than, up to, a maximum of → \le (filled circle, ray left);
• under, below, fewer than, less than → < (open circle, ray left).

Worked example. "A bouncy castle is for children who are over 3 years old and at most 12." Two conditions on the age a: "over 3" is strict, a > 3; "at most 12" is inclusive, a \le 12. Together:

On the number line: open circle at 3, filled circle at 12, shading between. A 3-year-old is turned away (open end); a 12-year-old bounces happily (filled end). And "23 kg luggage limit, strictly under"? That's m < 23 — but since a bag can't weigh a negative amount, the realistic picture is 0 \le m < 23: filled at 0, open at 23. Real-world context often quietly supplies a second wall.

Khan Academy walks through plotting an inequality on a number line here — a good second voice on the same idea: