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When working with numbers, it's essential to understand the concept of absolute value, which refers to the distance of a number from zero on the number line, without considering its direction. In essence, the absolute value gives us the magnitude of a number regardless of whether it is positive or negative.

For example, the absolute value of both \( -3 \) and \( 3 \) is \( 3 \), because each is three units away from zero. We denote the absolute value of a number \( x \) with vertical bars like this: \( |x| \). It's crucial to recognize that the absolute value of any number is never negative; it's either positive or zero.

Understanding this concept is particularly helpful when comparing negative numbers. If you're asked to compare \( -145.7183 \) and \( -14.57183 \), you first look at their absolute values, which are \( 145.7183 \) and \( 14.57183 \), respectively. From these, we can determine the larger absolute value, aiding us in accurately comparing the original negative numbers.

Inequality symbols are the mathematical shorthand used to compare the size or order of numbers. The most basic symbols are \( < \) and \( > \), which mean 'less than' and 'greater than', respectively. These symbols form the foundation of how we can express the relationship between numbers in terms of their size.

Using the previous example of comparing \( -145.7183 \) and \( -14.57183 \), once we've determined which absolute value is larger, we use inequality symbols to create a true statement about their relationship. We say that \( -145.7183 \) is 'less than' \( -14.57183 \) by placing the less than symbol between them: \( -145.7183 < -14.57183 \).

It's important to remember that when dealing with negative numbers, the number with the larger absolute value is actually the smaller number because it is further away from zero on the number line. Therefore, the inequality symbol that we use between two negative numbers can seem counterintuitive at first.

Negative numbers are numbers less than zero, represented by a minus sign \( - \) in front of them. They are used to describe values on a scale that goes below zero, such as temperature below freezing or debts. In the context of negative numbers, the rules for comparing them differ from those of positive numbers.

One fundamental principle to remember is that a negative number with a larger absolute value is considered less than a negative number with a smaller absolute value. This might seem opposite to how positive numbers work, but it's based on their positions on the number line. The further left a number is from zero, the smaller it is.

In our example, \( -145.7183 \) is to the left of \( -14.57183 \), making it a smaller number, which is why the true statement is \( -145.7183 < -14.57183 \). It's worth noting that when negative numbers are involved, understanding their relationship requires flipping the inequality once you've considered their absolute values.