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The concept of a number line is foundational in understanding inequalities in mathematics. It's a visual representation that helps us compare numbers. Think of the number line as a straight, horizontal line with markers placed at regular intervals. Each marker corresponds to an integer, with positive numbers to the right of zero and negative numbers to the left.

The further left a number, the smaller its value; conversely, numbers further to the right have greater values. This is particularly useful when visualizing where negative numbers stand in relation to each other, as some students might get confused thinking that '-20' is greater than '-3' simply because 20 is a bigger number than 3. However, the number line clearly shows that '-20' is to the left of '-3', signifying that it is indeed the smaller number.

With the number line, visualizing and understanding complex mathematical concepts becomes easy, and comparing magnitudes of different numbers simply involves looking at their relative positions on this imagined line.

The term 'inequality' in mathematics refers to the relationship between two values that aren't equal. When comparing two numbers, inequalities help us to express which number is smaller, larger, or if we cannot precisely state their order. The most common inequality signs are '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to).

In the context of the provided exercise, we used the less than sign '<' to compare two negative numbers. Understanding how to correctly interpret these signs is crucial to correctly solving inequalities. It's essential to remember that with negative numbers, the one with the greater absolute value, or distance from zero on the number line, is actually the smaller number. This is sometimes counterintuitive, which is why practicing with a variety of inequality problems is beneficial for students to become comfortable with comparing different types of numbers.

Comparing negative numbers can be tricky because the usual rules for positive numbers are flipped. The key point to remember is that with negative numbers, the one closer to zero is always larger. This is where the number line becomes incredibly helpful. For example, if we compare -3 and -20, we note that -3 is closer to zero.

Even though 20 is a higher number than 3, when we add the negative sign, this indicates a position on the left side of zero on the number line, which represents a lesser value. Therefore, -3 is greater than -20. It's crucial for students to understand that when dealing with negatives, smaller or more negative does not mean a larger number, and that negative numbers actually get smaller as they move left on the number line, away from zero.

Having a firm grasp on this concept allows students to tackle more complex mathematical problems with confidence, ensuring they understand the logic behind the rules of comparing negative numbers.