Inequalities and the Number Line

An equation like x = 4 has exactly one answer. An inequality like x > 4 has infinitely many — 4.1, 5, 100, and every number beyond. So the answer is not a single value but a range, and the number line is how we show it. There are four symbols to know: greater than, less than, greater than or equal to, and less than or equal to. The 'or equal to' versions matter, because they decide whether the boundary number itself is included, and that is the difference between an open and a filled circle when you draw it.

Drawing an inequality takes two decisions. First, which way does the arrow point — right for greater than, left for less than. Second, is the circle open or filled. An open circle means the boundary number is not included, which is what you use for greater than and less than. A filled circle means it is included, which is what you use for the 'or equal to' versions. There is a memory hook built into the symbols themselves: the little line underneath ≤ and ≥ is the same line that fills in the circle.

Solving an inequality uses exactly the same moves as solving an equation. For 3x + 2 > 14, subtract 2 from both sides to get 3x > 12, then divide both sides by 3 to get x > 4. The symbol simply travels along unchanged. Checking is slightly different though: instead of substituting one value, test a number inside your range and one outside it. Try x = 5, which gives 17 — greater than 14 ✓. Then try x = 3, which gives 11 — not greater than 14, exactly as it should be.

There is exactly one rule that makes inequalities different from equations: if you multiply or divide both sides by a negative number, you must flip the inequality sign. You can see why from a true statement — 3 is less than 5, but multiplying both by −1 gives −3 and −5, and −3 is now the *greater* of the two. So for −2x > 10, dividing by −2 gives x < −5, with the sign turned round. Test it: x = −6 gives 12, which is greater than 10 ✓, while x = 0 gives 0, which is not ✓. Note this applies only to multiplying and dividing — adding or subtracting a negative changes nothing.

The hardest part in practice is turning words into the right symbol. 'At least 18' means 18 or more, so x ≥ 18. 'At most 50' means 50 or fewer, so x ≤ 50. 'More than 12' excludes 12, so x > 12, and 'under $30' excludes 30. 'No more than 4' is x ≤ 4, and 'a minimum of 6' is x ≥ 6. The rule of thumb: 'at least', 'at most', 'no more than' and 'minimum' all include the boundary, while 'more than', 'under', 'over' and 'less than' exclude it. And in real contexts remember the answer is often a whole number — a lift with room for x more people cannot take 5.5 of them.