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Interval notation provides a concise way of describing a range of numbers on the real number line. It uses parentheses \( ( \) and brackets \( [ \) to indicate whether endpoints are included or excluded in the interval. For instance, the interval \( (-1,2] \) consists of all numbers greater than -1 and up to, and including, 2. The parenthesis \( ( \) signifies that -1 is not part of the interval, while the bracket \( ] \) indicates that 2 is included.

This notation is highly efficient because it can quickly convey both the bounds of an interval and the 'openness' or 'closedness' of these bounds. Open intervals exclude their endpoints, while closed intervals include them. A closed interval includes both endpoints, denoted by brackets \( [a, b] \) for the ends a and b. Conversely, an open interval \( (a, b) \) would indicate that neither a nor b are included. Mixed intervals like \( (-1,2] \) contain one endpoint but not the other.

Intervals represent subsets of the real number line that can be either bounded or unbounded. Bounded intervals have both upper and lower limits, such as \( (-1,2] \) where the limits are -1 and 2. In essence, the values in a bounded interval are contained within a particular range, and they do not extend to infinity.

On the other hand, unbounded intervals extend indefinitely in one or both directions. For example, \( (-\infty, 3) \) is unbounded to the left, meaning there is no lower limit to the values included in the interval. Similarly, \( (5, +\infty) \) is unbounded to the right. An interval can also be doubly unbounded, represented as \( (-\infty, +\infty) \) which includes all real numbers. When an interval is bounded, as in \( (-1,2] \) from the exercise, it indicates that there is a finite set of numbers contained within the interval.

Sketching a number line is an essential tool to visually represent intervals of real numbers. To sketch the interval \( (-1,2] \) on a number line, start by drawing a straight horizontal line. Mark a point for each boundary of the interval, labeling them appropriately. Since the interval \( (-1,2] \) includes -1 as a boundary but does not include it as a value, an open circle is drawn at -1. Conversely, 2 is part of the interval, indicated by a closed circle filled in at 2.

Once you've marked the endpoints, draw a solid line to connect them, indicating that all numbers between those points are included in the interval (except for the number -1 itself in this case). Sketching number lines this way provides a clear picture of which numbers are part of the interval and how it relates to the entire set of real numbers.

Inequalities are statements about the relative size or order of two values. They are a fundamental part of describing intervals in mathematics. The inequality \( x > -1 \) tells us that x is greater than -1, and \( x \leq 2 \) indicates that x is less than or equal to 2. These two inequalities can be combined to describe the interval \( (-1,2] \) concisely.

When working with inequalities, we often represent them on a number line to provide a visual understanding. Inequalities help us identify whether an interval is open or closed and what values are included in it. It's crucial to remember that the solution to an inequality is not just a single number, but rather a range of numbers that satisfy the inequality, which is what an interval represents.