An inequality is like an equation, but instead of "equals" it says one side is bigger or smaller than the other. Real life is full of them: you must be at least 12 to see a film, a lift holds no more than 8 people, a saver wants more than £100 in the bank. The four symbols are < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to).
Why this matters: most real situations aren't a single exact answer but a range of allowed values — budgets, speed limits, safe doses, pass marks. Inequalities let you describe and solve those ranges, and they lead straight into graphing regions and linear programming later on.
x < 5: open circle (5 not included), shaded everywhere to the left
Reading the symbols
A handy trick: the symbol is a hungry mouth that always opens towards the bigger number. So 3 < 5 and 5 > 3 say the same thing. Adding the line underneath (≤ or ≥) means "or equal to" is allowed as well.
The wide open end always points at the larger value: 3 < 5.
Symbol
Means
Example in words
<
less than
x < 5: below 5, but not 5 itself
>
greater than
x > 5: above 5, but not 5
≤
less than or equal to
x ≤ 5: 5 and everything below
≥
greater than or equal to
x ≥ 5: 5 and everything above
Solving them — almost like equations
You solve an inequality the same way as an equation: do the same to both sides to peel it apart. The only difference is you keep the inequality symbol instead of an equals sign — and the answer is a range of values, not a single one.
3x + 1 < 16: subtract 1 → 3x < 15, divide by 3 → x < 5 (x can be any number below 5).
The one big rule: if you multiply or divide both sides by a negative number, you must flip the symbol. So −2x ≥ 6 → divide by −2 and flip → x ≤ −3. (Sense-check with numbers: 2 < 4 is true, but multiply by −1 and −2 < −4 is false — you must flip to −2 > −4.)
Multiplying by −1 reverses the order, so the symbol must flip.
Showing it on a number line
Mark the boundary value, then shade the whole range that works:
Open circle ○ for < or > — the number itself is not included.
Filled circle ● for ≤ or ≥ — the number is included.
List the integers where −2 < x ≤ 3: −2 is excluded (open) but 3 is included (filled), so the whole numbers are −1, 0, 1, 2, 3.
Open at −2 (excluded), filled at 3 (included): integers −1, 0, 1, 2, 3.
In a nutshell: solve just like an equation and keep the symbol — the mouth eats the bigger number, a hollow circle excludes and a filled one includes, and you flip the symbol only when you multiply or divide by a negative.