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When it comes to mathematics, the number line is basic yet instrumental in visually representing numbers and their relationships. It is essentially a straight line with numbers placed at intervals that demonstrate magnitude and direction. Numbers to the left represent lower values, while numbers to the right indicate higher values.

For negative integers, this can be a bit counterintuitive at first, since they are represented on the left side of the zero (which is the midpoint of the number line), with larger negative numbers placed further left. This means that although -3 might appear 'less' than -13, its position to the right of -13 on the number line actually makes it greater. In a number line, moving to the right, no matter whether from negative to less negative, or from positive to more positive, always means increasing value.

The concept of inequalities shines when it comes to comparing numbers. An inequality states that two values are not equal and that one is greater (>) or lesser (<) than the other. Understanding this concept with positive numbers is straightforward, but negativity often brings confusion.

To demystify, consider this: each negative number is less than zero, but among negative numbers, those with a smaller absolute value—or distance from zero—are actually 'greater.' This seems paradoxical, but it simply means that -2 is greater than -4 because it's closer to zero. Therein lies the importance of viewing negative numbers as points on the number line, making it easier to see which are larger or smaller comparative to one another.

To dive a bit deeper into comparing negative integers, let's go back to the basics. An integer is a whole number and can be positive, negative, or zero. The negative integers are to the left of zero on the number line and become smaller the further they are from zero.

When asked to compare negative integers, like in the exercise, where -3 and -13 are compared, you should think in terms of distance from zero. In this case, -3 is only three units away from zero, while -13 is thirteen units away, making -3 the larger number because it is 'less negative.' It's a bit like thinking who owes less money in debts; owing 3 dollars (-3) is definitely a better situation than owing 13 dollars (-13).