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Inequalities are a fundamental concept in mathematics, representing the relative size or order of two numbers. The symbols most commonly used are '<' to denote less than and '>' for greater than. It's a way of comparing numbers to see if one is larger, smaller, or if they're not equal at all.
When we express an inequality such as 'a < b', it means that 'a' is smaller than 'b'. Similarly, 'a > b' indicates that 'a' is larger than 'b'. It's essential to note that these symbols always point to the smaller number, resembling a pair of open jaws ready to eat the larger value.
Understanding these symbols and their proper usage is crucial for solving inequality problems. When evaluating an inequality, it's important to imagine the number line, where numbers to the right are greater, and to the left are smaller. With negative numbers, this can become slightly tricky because we are used to thinking of larger numbers as being 'more,' but in the negative zone, larger means more negative, hence actually being 'less' in value.

Dealing with negative numbers can be counterintuitive since the more negative a number is, the smaller its value is considered. When comparing negative numbers, it's crucial to recognize that they are 'less' than positive numbers and that among the negatives, the number closer to zero is 'greater.'
For example, -3 is greater than -5 because -3 is closer to zero on the number line. This might seem confusing because -3 is technically 'less negative' than -5, which is why it's actually greater in value. To properly compare negative numbers, envision them on the number line; the number further to the right (towards the positive side) is always going to be the greater number.

Mathematical symbols are the shorthand of math; they convey complex relationships between numbers and operations with concise clarity. Apart from the inequality signs, other foundational symbols include '=', which denotes equality, and '≠' for not equal. There are also '≤' and '≥', standing for less than or equal to, and greater than or equal to, respectively.
Being proficient with these symbols is more than memory; it’s understanding their meaning and application in various mathematical contexts. They help streamline communication and problem-solving by providing a universal language for mathematicians. In inequalities, clarity in understanding these symbols is paramount as they guide the logic of the comparison being made and ensure accurate solutions.