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Interval notation is a mathematical system for describing a range of numbers. It is incredibly efficient for expressing intervals on the real number line, and it is commonly used when dealing with continuous data. A fundamental aspect to understand is the use of brackets and parentheses: square brackets \[ \] indicate that the endpoint is included in the interval, known as a closed interval, while parentheses \( \) suggest that the endpoint is not included, known as an open interval.

For example, the interval notation \[ -3, 1 \] from the exercise represents all real numbers from -3 to 1, including -3 and 1 themselves. In contrast, \( -3, 1 \) would signify that -3 and 1 are not part of the set. Additionally, combinations like \[ -3, 1 \) or \(-3, 1 \] are also possible, where one endpoint is included, and the other is not, called half-open intervals or half-closed intervals.

A number line graph is a visual representation of numbers on a straight line, where each point corresponds to a number. This tool is essential for illustrating intervals and inequalities. To graph an interval such as \[ -3, 1 \], start by drawing a horizontal line and marking evenly spaced points that represent numbers, ensuring that the numbers of interest are included.

Following our example, you plot a point at -3 and another at 1. Since it is a closed interval, you'd draw solid dots on these points to indicate that they are part of the interval. Then, shade the line segment between -3 and 1 to show that every number in this range is included in the interval. A number line graph provides a straightforward visual of which numbers satisfy the interval or inequality being considered.

Inequalities express the relation between two values with regard to their size, indicating that one is larger or smaller than the other. Common inequality signs include < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Inequalities are crucial when defining ranges of numbers, like in set-builder notation.

The set-builder notation \( \{ x \, \vert \, -3 \leq x \leq 1 \} \) is based on an inequality. It establishes that the numbers x must be greater than or equal to -3 and less than or equal to 1 to be part of the set. This range can be easily visualized on a number line, where the solution to the inequality \( -3 \leq x \leq 1 \) is marked. Inequalities often come in systems, where multiple conditions must be met simultaneously, providing a foundational concept for understanding both basic and complex mathematical concepts.