Inequalities and Absolute Value
Absolute Value: Definition and Properties
Absolute value as distance from zero, the two-case definition, the identity |a| = √a2, and the triangle inequality.
What this page covers
- State the two-case definition and apply it
- Interpret absolute value as distance on the number line
- Use the multiplicative and reflection properties
- State the triangle inequality and say when it is an equality
The definition
The second line is the one that causes hesitation. When a is negative, -a is positive, so the rule returns a positive value as it should. At a = -3 it gives -(-3) = 3.
Informally, absolute value strips the sign. The source puts it more usefully:
|a| is the distance of a from 0 on the number line.
Both points are marked by their distance from zero: 3.7 to the left, 4.9 to the right. Distance is never negative, which is why |a| ≥ 0 always.
The distance reading is the one that makes the equations and inequalities easy. |x| = 3 asks which points are 3 away from zero; |x| < 3 asks which are closer than 3.
The square-root identity
Squaring destroys the sign and the principal square root returns the non-negative value, so the composition does exactly what the two-case definition does — without the cases.
| a | a2 | √a2 | |a| |
|---|---|---|---|
| 3 | 9 | 3 | 3 |
| -3 | 9 | 3 | 3 |
| 0 | 0 | 0 | 0 |
This identity is why absolute values appear unexpectedly when even roots undo even powers. √x2 = |x|, and more generally (x6)1/2 = |x3| — a point the source makes in Week 2 and which is developed in Fractional Exponents and Radicals.
The properties
| Property | Name | Reading |
|---|---|---|
| |ab| = |a||b| | Multiplicative | Absolute value distributes over multiplication exactly |
| |-a| = |a| | Reflection | A point and its mirror image are the same distance from zero |
| |a + b| ≤ |a| + |b| | Triangle inequality | Over addition it distributes only as an inequality |
| |ab| = |a||b| | Quotient | Follows from the multiplicative property, b ≠ 0 |
Multiplication is exact; addition is not. |a + b| is generally less than |a| + |b|, and only equal when a and b point the same way.
The triangle inequality
The source demonstrates both cases with the same two numbers, which makes the condition for equality plain.
Equality — same signs
Both numbers are positive, so nothing cancels and the two sides agree.
Strict inequality — opposite signs
The signs oppose, so partial cancellation occurs inside the absolute value but not outside it. The gap between the two sides measures how much cancelled.
|a + b| = |a| + |b| exactly when a and b have the same sign, or at least one is zero. Otherwise the inequality is strict.
The name comes from geometry: in a triangle no side exceeds the sum of the other two, with equality only when the triangle degenerates to a straight line. The same statement governs vector magnitudes, where |u + v| ≤ |u| + |v| with equality when the vectors are parallel and point the same way.
Distance between two points
Absolute value measures distance from zero. Distance between any two points follows immediately.
This reading turns absolute-value statements into geometric ones, which is usually the fastest route to a solution.
| Statement | Geometric reading | Solution |
|---|---|---|
| |x| = 3 | x is 3 from 0 | x = ±3 |
| |x - 5| = 2 | x is 2 from 5 | x = 3 or x = 7 |
| |x + 5| = 3 | x is 3 from -5 | x = -2 or x = -8 |
| |x - 5| < 2 | x is within 2 of 5 | (3, 7) |
| |x - 5| > 2 | x is further than 2 from 5 | (-∞, 3) ∪ (7, ∞) |
The source works |x + 5| = 3 this way, marking -2 and -8 on a line with arrows of length 5 and 3. Reading x + 5 as x - (-5) is the step that makes the centre visible.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| |-a| = -a always | |-a| = |a| | At a = 3: |-3| = 3, not -3 |
| |a + b| = |a| + |b| | ≤, with equality only for matching signs | |5 - 7| = 2 ≠ 12 |
| √x2 = x | |x| | At x = -3 the left is 3 |
| Reading |x + 5| as distance from 5 | It is distance from -5 | x + 5 = x - (-5) |
| Solving |x| = -3 | No solution | A distance is never negative |
| |ab| ≠ |a||b| | It is equal | Multiplication is exact; only addition degrades to ≤ |
Frequently asked questions
Why is |a| = -a when a is negative?
Because -a is then positive. If a = -3 then -a = 3, which is the distance from zero. The minus sign is not making the answer negative; it is undoing an existing negative.
Is |a| = √a2 really an identity?
Yes, and it is often the most convenient definition. Squaring discards the sign and the principal square root returns a non-negative value, which is exactly what absolute value does.
When is |a + b| = |a| + |b|?
Exactly when a and b have the same sign, or one is zero. The source illustrates both cases: |5 + 7| = 12 = |5| + |7|, but |5 - 7| = 2 while |5| + |-7| = 12.
Does absolute value distribute over multiplication?
Yes: |ab| = |a||b| always. It is addition where it fails, which is the content of the triangle inequality.
Source. Handwritten teaching notes, Week 4, pages 2-3.
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.