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Understanding decimal place value is essential when working with decimals. A decimal number is composed of a whole number part and a fractional part, separated by a decimal point. The place value of digits after the decimal point is determined by their position. Immediately to the right of the decimal point is the tenths place, followed by the hundredths, thousandths, and so on.

When comparing two decimals, look at the highest place value first. In the pair of decimals 121.532 and 121.503, both share the same whole number part, 121, and hence we must compare the tenths. When the tenths are the same, as they are here (both are 5), we move on to compare the hundredths place, which is the second digit after the decimal point. The number with the higher value in this digit will be the greater decimal. In our example, the hundredths place is 3 in 121.532 and 0 in 121.503, indicating that 121.532 has a greater value.

Mathematical inequalities help us show the relative size of two numbers. The symbols used are < for 'less than', > for 'greater than', and = for 'equal to'. To determine which inequality symbol to use when comparing two decimals, we must first compare their values as described in the decimal comparison process.

Using inequalities allows us to express the relationship between numbers quickly and is especially useful when dealing with variables and algebraic expressions. However, in the context of decimal comparison, once we determine which decimal has the larger place value at the first point of difference, we can confidently use the appropriate inequality symbol to show this relationship.

The process of decimal comparison involves analyzing the digits' value from left to right after ensuring the whole number parts are identical. If the whole number parts differ, the comparison is straightforward as the number with the larger whole number part is greater.

If the whole numbers are the same, as seen with 121.532 and 121.503, we compare each place value starting from the tenths place. In this example, we can see that the numbers differ at the hundredths place. Since 3 is greater than 0, we say 121.532 is greater than 121.503, which we can represent with the inequality 121.532 > 121.503.

Remember that precision is key when comparing decimal numbers. Even a small difference in a lower place value can determine the inequality, which is a crucial concept that students must grasp for accurate comparison.