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Inequalities are mathematical expressions that compare two values, typically by using symbols such as <, >, \( \leq \) (less than or equal to), and \( \geq \) (greater than or equal to). These symbols help us understand the relationship between numbers and the range of values that can satisfy a condition.

In the given exercise, \( -2 < x \leq 5 \) represents an inequality showing all the numbers that are greater than -2 but less than or equal to 5. To solve such problems, we need to consider the set of numbers that fit within this range. It's important to note that the number -2 isn't included, while 5 is included in the solution set, as indicated by the symbols. Understanding how to read and interpret these symbols is crucial for graphing the inequality correctly on a number line.

A number line is an essential tool in mathematics for visually representing and comparing numbers. It's a straight line with points that are placed at equal intervals. Each point corresponds to a number.

To graph an inequality on a number line, as showcased in our exercise, you start by drawing this line and marking the limits defined by the inequality. Zero is customarily placed at the center to provide a clear reference for positive and negative numbers. The step-by-step solution suggests that the points -2 and 5 should be plotted adequately apart to visually represent the interval with precision. Once these limits are on the line, the range of the inequality is indicated by highlighting or shading the area between these two points, taking care to symbolize whether a limit is included in the set.

Interval notation is a compact way of writing the set of all numbers between two endpoints. It involves using brackets and parentheses to communicate which endpoints are included or excluded from the interval.

In interval notation, a square bracket \( [ \) or \( ] \) means that the endpoint is part of the interval, while a round bracket \( ( \) or \( ) \) indicates that the endpoint is not part of the interval. For our exercise, the set \( \{x \mid -2 < x \leq 5\} \) would be written in interval notation as \( (-2, 5] \)—the parenthesis around -2 means that -2 is not included (just as the open circle on the number line does), and the square bracket around 5 indicates that 5 is included (like the filled circle). Learning interval notation is a concise method to represent intervals and is widely used in mathematics.