Triangle Inequality

The triangle inequality theorem states that the absolute value of a sum is always less than or equal to the sum of the absolute values: $$ |a + b| \le |a| + |b| $$ for all real numbers a and b in ℝ.

This fundamental property is also widely used in geometry, where it underpins the principle that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side (see the triangle inequality theorem).

A Practical Example

Let’s look at an example with two numbers: a = 2 and b = - 3.

The absolute value of their sum is one:

$$ |2 + (-3)| = 1 $$

The sum of their absolute values is five:

$$ |2| + |-3| = 5 $$

Therefore, the triangle inequality holds true:

$$ |2 + (-3)| \le |2| + |-3| $$

$$ |-1| \le 2 + 3 $$

$$ 1 \le 5 $$

And so on.