What it is
An inequality compares two expressions with <, ≤, > or ≥. Unlike an equation, it usually has not one answer but many. x < 3 holds for 2, for 0, for −100 and for 2.999, but not for 3.
The set of all numbers that satisfy an inequality can be written as an interval:
| Inequality | Interval | End points |
|---|---|---|
| 2 ≤ x < 5 | [2, 5) | 2 is in, 5 is not |
| x > 3 | (3, ∞) | 3 is not in |
| x ≤ −1 | (−∞, −1] | −1 is in |
| −2 < x < 2 | (−2, 2) | neither end point |
A square bracket means the end point is included (the interval is closed at that end) and a round bracket that it is not (open). Infinity ∞ is not a point on the number line, so that end is always open.
A few inequalities solved:
- 2x + 1 < 7 gives 2x < 6 and x < 3. Check: x = 2 gives 5 < 7, true, and x = 3 gives 7 < 7, false.
- −2x > 6 gives, dividing by −2 and turning the sign, x < −3. Check: x = −4 gives 8 > 6, true, and x = −3 gives 6 > 6, false.
- |x| < 2 means x is less than 2 from zero: −2 < x < 2, the interval (−2, 2).
x = 3 is not an inequality but an equation with one answer. And 5 ≤ x ≤ 2 is an inequality with no solution: no number is both at least 5 and at most 2, so it describes no interval at all.
Why it matters
Numbers that meet a condition turn up everywhere: “at least 18 years”, “at most 120 km/h”, “between 0 and 100 per cent”. The inequality is the mathematical language for the condition and the interval for the answer.
- How to solve an inequality, and when the sign turns, is under rules for inequalities.
- Absolute value expresses distance, and inequalities with it, like |x| < 2, become intervals.
- An interval is a set: [2, 5) is the x among the real numbers for which 2 ≤ x < 5.
The equation solver solves inequalities and writes the answer both as an inequality and as an interval.
Common misconceptions
- “≤ and < are the same.” They differ in exactly one point. x ≤ 3 contains 3, x < 3 does not.
- “Square and round brackets are just style.” [2, 5) and [2, 5] have different contents: 5 is in the second.
- “You can write ∞ with a square bracket.” Infinity is not a number and is never included, so (3, ∞) is right and (3, ∞] is wrong.
- “The sign turns with every operation.” It turns only when you multiply or divide by a negative number. Adding or subtracting does not change it.
- “An interval holds only a few numbers.” Between 2 and 3 there are infinitely many, such as 2.5, 2.01 and 2.999.
Common questions
What does x ∈ ℝ mean?
That x is a real number, a number on the number line. Writing x ∈ [2, 5) means x lies in the interval,
that is 2 ≤ x < 5.
How do you write “all numbers except 3”?
It is not one interval but two: (−∞, 3) and (3, ∞), joined with the union sign: (−∞, 3) ∪ (3, ∞). See
set notation.