Triangle Inequality for Absolute Values

Triangle Inequality

The triangle inequality is a mathematical principle that is used all over mathematics. The way the triangle inequality is used most is in geometry. The triangle inequality states that: For any triangle the length of any two sides of the triangle must be equal to or greater than the third side. Sometimes seen as: $$|X+Y|\le|X|+|Y|$$ The inequality works not only if X and Y are both real numbers (scalars), but also if X and Y are vectors (of the same dimension).

From the above diagram you're able to see that if the sum of $X$ and $Y$ was shorter than $Z$ then $X$ and $Y$ would be unable to connect at a vertex and the shape would no longer be a triangle.

There are a few different ways you can write the triangle inequality. Any way you choose is correct but certain ways are useful in describing certain situations. Lets start off with the original triangle inequality: $$|X+Y|\le|X|+|Y|$$ Now lets think what would happen if let $X=A$ and $Y=-B$. By plugging our new values into the original equation we get: $$|A+(-B)|\le|A|+|-B|$$ Which is equal to $$|A-B|\le|A|+|B|$$ Another way to phrase $|A-B|$ is: The distance between points $A$ and $B$. So what the above version of the triangle inequality is saying that the distance between the points $A$ and $B$ will always be shorter than or equal to the distance from $A$ to $0$ plus the distance from $B$ to $0$.

Another way to write the triangle inequality is as follows: Let $X=(A-B)$ and $Y=(B-C)$. By plugging into the original triangle inequality we get: $$|(A-B)+(B-C)|\le|(A-B)|+|(B-C)|$$ By simplifying that inequality we are able to get: $$|A-C|\le|A-B|+|B-C|$$ In other words what the above inequality is saying is that the distance between the points $A$ and $C$ will always be less than or equal to the distance from $A$ to $B$ and then the distance from $B$ to $C$. Which makes logical sense because you can't go to some point $B$ in order to shorten the distance from $A$ to $C$.

Examples with Triangle Inequality $\\$

The triangle inequality is a theorem a theorem about distances. $$\left|a+b\right|\le\left|a\right|+\left|b\right|$$

Note that we are taking the absolute values of slightly different things on the two sides. On one side, we are taking the absolute value of the sum; on the other, we are taking the sum of the absolute value.

The formula holds for all real numbers.

Try on negative numbers

Test the formula with $a=-5$ and $b=3$. $$\left|a+b\right|=\left|-5+3\right|=\left|-2\right|=2$$ $$\left|a\right|+\left|b\right|=\left|-5\right|+\left|3\right|=5+3=8$$

We can see that $\left|a+b\right|\le\left|a\right|+\left|b\right|$.