Your phone plan caps how much data you can use; a lift has a "maximum 8 people" sign; an online shop drops the delivery charge once you spend enough. Everyday life is full of limits, where a whole range of answers is allowed rather than one exact number — and writing and solving those limits is exactly what an inequality is for.

An equation like 3x + 2 = 11 has exactly one answer: x = 3, and nothing else fits. But change that = into a > and everything opens up. Ask instead 3x + 2 > 11 and there is no single answer — there is a whole range of them.

Try x = 4: 3(4) + 2 = 14, which really is bigger than 11. Try x = 100: also works. Try x = 3.01: works. In fact any number bigger than 3 works, and no number 3 or below does. The solution isn't a point — it's everything to the right of a line in the sand:

Here's the good news: you solve an inequality almost exactly like an equation — the same operations to both sides. There is just one twist that trips up nearly everybody, and we'll make sure it never trips up you.

A linear inequality like 2x + 1 > 7 is solved almost exactly like an equation: you peel away the operations around x by doing the inverse operation to both sides, until x stands alone. If you can already solve a two-step equation, you can already do most of this.

Subtract 1 from both sides, then divide both sides by 2:

The answer isn't a single number — it's every number bigger than 3, the whole solution set you can picture on a number line.

Drawing the answer: open and filled circles

Because the answer is a range, we picture it on a number line. Mark the boundary value, then shade an arrow across every number that works. There is a neat convention for the boundary itself:

• An open circle (a hollow ring) for < or > — the boundary is not included. For x > 3, the number 3 itself does not satisfy it (3 is not bigger than 3), so we leave the circle hollow.
• A filled circle (a solid dot) for \le or \ge — the boundary is included. For x \ge 3, the number 3 does count, so the dot is filled in.

Here is x > 3 drawn out — a hollow ring at 3 and an arrow running off to the right, sweeping up every number bigger than 3. This is the same picture you built when you met inequalities on a number line.

Worked example: a two-sided inequality

Sometimes x is trapped between two values, written as a double inequality. The rule is simple: whatever you do, do it to all three parts at once. Solve:

Divide every part by 2 (a positive number, so no sign flips):

So x is any number greater than -1 and up to and including 3. On a number line that's an open circle at -1, a filled circle at 3, and a bar joining them. Whole numbers that fit: 0, 1, 2, 3 — but not -1 (open) and not 4.

The one extra rule: flipping the sign

Inequalities behave just like equations under adding and subtracting, and under multiplying or dividing by a positive number. There is exactly one difference:

If you multiply or divide both sides by a negative number, flip the inequality sign.

Take 4 > 2 (true) and divide both sides by -2. Without flipping you'd get -2 > -1, which is false — but -2 < -1 is true. Negating both sides reverses their order, just like reflecting points across 0 on the line (this is the same mirror you meet with negative numbers). So the rule keeps the statement true:

Notice: dividing by -2 turned > into <. Check it — x = -4 gives -2(-4) = 8 > 6: yes, -4 < -3 is in the solution. It really is the numbers below -3.

See it solved

Step through the solution of 2x + 1 > 7 one move at a time — each line is the inverse operation applied to both sides — then watch the final step show what happens when the divisor is negative and the sign must flip.

See it explained

Sal Khan works through solving a two-step inequality, including when to reverse the sign.