Properties of absolute values:

1. Non-negativity: The absolute value of any real number x is always greater than or equal to zero. No matter if x is positive, negative, or zero, the absolute value will never be negative. It represents distance, and distance cannot be negative.
   • | 3 | = 3
   • | -3 | = 3
   • | 0 | = 0
2. Identity: The absolute value of x is x itself if x is greater than or equal to zero. When a number is already positive or zero, its distance from zero is just the number itself.
   • If X = 5, then | 5 | = 5
   • if X = 0, then | 0 | = 0
3. Symmetry: The absolute value of −x is the same as the absolute value of x. Both positive and negative numbers are equally far from zero. For example, 5 steps to the right and 5 steps to the left both cover a distance of 5 units.
   • | -5 | = 5
   • | 5 | = 5
4. Triangle Inequality: The absolute value of the sum of two numbers x and y is less than or equal to the sum of their absolute values. This property tells us that the direct route (adding x and y first) is never longer than taking each step separately and then adding the distances. It essentially states that going directly to your destination is the shortest path.
   • | 3 + (- 4) | = | - 1 | = 1
   • | 3 + | - 4 | = 3 + 4 = 7
   • | 3 + (- 4) | ≤ | 3| + | - 4| = 1 ≤ 7
5. Multiplicative Property: The absolute value of the product of two numbers x and y is equal to the product of their absolute values. When you multiply two numbers, whether they are positive or negative, the distance from zero (absolute value) of the product is the same as multiplying their individual distances.
   • | 3 x -2 |= | - 6| = 6
   • |3| x |-2| = 3 x 2 = 6