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Understanding how to compare numbers is a fundamental skill in mathematics. It helps us make sense of numerical value and magnitude. For instance, when we consider two numbers, such as 12 and 19, our goal is to determine which number is greater or if they are equal. In the comparison of 12 to 19, we observe that each digit from left to right has a certain place value, with digits further to the left having a higher value. In this case, since both numbers are two digits long, we compare the leading digits first: the 1 in 12 and the 1 in 19 are equal; however, moving to the next digits, the 2 in 12 is smaller than the 9 in 19. Hence, we deduce that 12 is less than 19.

Comparing numbers leads to understanding their relative sizes and positions on a number line. When multiple numbers are compared, it helps in arranging them in ascending (increasing) or descending (decreasing) order. This foundational concept is vital in math because it sets the stage for more advanced topics like working with inequalities, which we use to express the relative size of numbers.

Inequality symbols are at the core of expressing mathematical relationships between numbers that are not equal. The two primary symbols we encounter are the less than symbol (<) and the greater than symbol (>). These symbols are visual representations that allow us to express the inequality between two values succinctly. For example, when comparing the numbers 12 and 19, since 12 is the smaller number, we use the less than symbol, denoted by <, to illustrate the relationship: \(12 < 19\).

But inequalities don't end there. There are also symbols to represent 'less than or equal to' (\(\) and 'greater than or equal to' (\(\) These encompass situations where the numbers might be equal or have the characteristics of the basic inequality symbols. Inequalities are pivotal in effectively communicating a range of possibilities, such as in solving equations or representing real-world scenarios where values can vary within certain limits.

Writing inequalities allows us to express the relationship between numbers and variables beyond simple equality. When constructing an inequality, we begin by analyzing the values we wish to compare. As seen in the exercise comparing 12 and 19, we identified that 12 is smaller than 19, leading us to choose the correct inequality symbol. Therefore, we write the true statement as \(12 < 19\).

Writing inequalities involves deciding not only on the correct direction of the inequality symbol but also understanding what the inequality represents. For example, if a student scores 85 on a test and the passing grade is 60, we can express this as \(85 > 60\), which shows the student’s score is greater than the passing score. Similarly, when setting up equations or conditions in word problems, properly written inequalities guide the solution process. For instance, finding all numbers greater than 12 would be expressed as \(n > 12\), with 'n' being any number larger than 12. This clear, concise notation is critical in mathematics to establish constraints and parameters within a problem.