Prove the absolute value equation |a-5| = |5-a|

Published on October 27, 2024

Algebra

The equation |a - 5| = |5 - a| is proven by considering two cases: when (a - 5) is greater than or equal to zero and when (a - 5) is less than zero. In both cases, the equation holds true, demonstrating that the absolute value of the difference between a and 5 is equal to the absolute value of the difference between 5 and a for all values of a.

Question

prove |a-5|=|5-a|

Answer

Let’s break down how to prove the equation

Understanding Absolute Value

The absolute value of a number is its distance from zero, regardless of direction. This means:

If a number is positive or zero, its absolute value is itself. For example, |3| = 3 and |0| = 0.
If a number is negative, its absolute value is its positive counterpart. For example, |-3| = 3.

Two Cases

Because of how absolute value works, we need to consider two cases:

Case 1: (a - 5) is greater than or equal to zero