What inequalities are

An inequality compares two numbers or expressions with one of four signs:

• < (less than) and > (greater than) are strict: equality is not allowed.
• ≤ (less than or equal to) and ≥ (greater than or equal to) are slack: equality is allowed.

A linear inequality in one variable looks like ax + b < 0 (or with >, ≤, ≥), with a ≠ 0. A solution is any value of x that makes it true.

Here is the big difference from equations. x + 5 = 4 has exactly one solution, x = –1. But x + 5 > 4 is true for x = 0, x = 1, x = 1.1, x = 100 ...: it has infinitely many solutions, all the numbers greater than –1. Some inequalities have no solution at all.

We solve inequalities almost like equations, with two rules:

• Rule 1: you may add or subtract the same number on both sides (move a term across with its sign changed). The inequality sign stays.
• Rule 2: you may multiply or divide both sides by the same number. If it is positive, the sign stays; if it is negative, the sign reverses.

Why reverse? Start from 3 > 2, which is true, and multiply both sides by –1. You get –3 and –2, and on the number line –3 is to the left of –2, so –3 < –2. Keeping '>' would be false.

Examples:

• 3x + 7 > 1: subtract 7, 3x > –6; divide by 3 (positive), x > –2.
• –4x ≥ 20: divide by –4 (negative) and reverse, x ≤ –5.

The answer also depends on which numbers x may be:

• 5x < 24 gives x < 4.8. If x must be a natural number, the solution set is {1, 2, 3, 4}. If x may be any integer, it is {..., –1, 0, 1, 2, 3, 4}. If x is real, it is the interval (–∞, 4.8).
• x ≤ –5 has no natural-number solutions at all, but in the integers it is {..., –7, –6, –5}.

Interval notation is handy for real solutions: a round bracket leaves the end out and a square one keeps it, and ∞ always takes a round bracket. So x ≤ –5 is (–∞, –5], and x > –2 is (–2, ∞).