Inequalities and Absolute Value
Inequalities and Interval Notation
Order on the real line, the two notations for describing an interval, and the one rule that makes solving an inequality different from solving an equation.
What this page covers
- Read and write interval and set notation for a region of the line
- Solve a linear inequality, reversing the sign where required
- Represent a solution on a number line with open and closed endpoints
- Say why inequalities are a real-number topic only
Order on the real line
An inequality compares size. The source opens the topic with a restriction that is easy to read past and important to keep: inequalities apply only in the real numbers.
There is no way to order the complex numbers that respects the arithmetic. Asking whether i > 0 leads to a contradiction either way, so < is simply not defined there. Every inequality on this page is a statement about ℝ.
- a < b
- a is less than b
- b > a
- b is greater than a — the same statement
- a ≤ x
- a is less than or equal to x
- x ≥ a
- x is greater than or equal to a
- a < x < b
- x lies strictly between a and b
The double form a < x < b is an abbreviation for two statements at once: a < x and x < b. The source introduces it that way, which is the safe reading.
Interval notation
A solution is usually a stretch of the line rather than a handful of points, and interval notation names such a stretch compactly.
| Inequality | Interval | Endpoints | Meaning |
|---|---|---|---|
| a < x < b | (a, b) | Both open | Between, excluding both ends |
| a ≤ x ≤ b | [a, b] | Both closed | Between, including both ends |
| a ≤ x < b | [a, b) | Left closed, right open | Includes a, excludes b |
| a < x ≤ b | (a, b] | Left open, right closed | Excludes a, includes b |
| x > a | (a, ∞) | Open at a | Everything above a |
| x ≤ b | (-∞, b] | Closed at b | Everything up to and including b |
∞ is a direction, not a number, and no value equals it. [3, ∞) is correct; [3, ∞] is not. The bracket at the finite end is decided independently.
(1, 5] — hollow at 1 because it is excluded, solid at 5 because it is included.
Set notation says the same thing differently: {x : 1 < x ≤ 5}, read as the set of x such that. Both appear in the source, and both are worth being fluent in.
Solving linear inequalities
The method is that of linear equations with one exception, which the source sets out in capitals.
| Operation | Effect on the inequality |
|---|---|
| Add or subtract from both sides | Sign unchanged |
| Multiply or divide by a positive number | Sign unchanged |
| Multiply or divide by a negative number | Sign reverses |
The reason is geometric. Multiplying by -1 reflects the line through zero, and reflection reverses order. The source illustrates with the pair 3 < 4 and -3 > -4, which is as clear a demonstration as any.
3 < 4 on the right; reflecting gives -3 > -4 on the left. The order has flipped.
Worked example — no reversal
x > 4, that is (4, ∞).
Worked example — with reversal
Test x = -10: left 3(-10) - 4 = -34, right 5 + 4(-10) = -35, and -34 > -35. Correct. Test x = 0: -4 > 5 is false, as it should be.
Substituting one value from inside the claimed solution and one from outside it is a complete check on the direction of the sign, and takes seconds.
An applied example from the source
How long is the ball in the air?
The source models a ball thrown upward with height h = 495t - 4910t2 and asks for how long h > 0.
The product is positive when both factors share a sign. For t > 0, the first factor is positive, so the second must be too: 2 - t > 0, that is t < 2.
The lower bound t > 0 comes from the physical situation, not from the algebra. Negative time is not meaningful here, and stating that explicitly is part of the answer. The source writes t < 2; the full interval is (0, 2).
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Not reversing after dividing by a negative | The sign flips | Test one value from inside the answer |
| [3, ∞] | [3, ∞) | Infinity is never attained |
| Solid dot for a strict inequality | Hollow dot | < and > exclude the endpoint |
| Treating a < x < b as one statement to manipulate loosely | It is two statements | Split them if in any doubt |
| Multiplying by an expression of unknown sign | Split into cases, or use a sign diagram | x could be negative and flip the inequality |
| Applying inequalities to complex numbers | Not defined | Order exists on ℝ only |
Frequently asked questions
Why does multiplying by a negative reverse the sign?
Because multiplying by -1 reflects the line through zero, and reflection reverses order. 3 < 4 but -3 > -4. The source gives exactly this pair as the illustration.
When is an endpoint open and when closed?
Closed (a solid dot, a square bracket) when the endpoint itself satisfies the inequality, which happens with ≤ and ≥. Open (a hollow dot, a round bracket) with strict < and >.
Why is ∞ always given a round bracket?
Because it is not a number and cannot be attained. (3, ∞) is correct; [3, ∞] is not, whatever the other endpoint does.
Do inequalities work for complex numbers?
No. The source states that inequalities apply only in the real numbers. There is no ordering of the complex numbers compatible with the arithmetic.
Source. Handwritten teaching notes, Week 3, pages 9-11.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.