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A number line is a simple tool that helps visualize numbers and understand relationships between them. Imagine a horizontal line that stretches infinitely in both directions. The line is marked with numbers at regular intervals, allowing you to see every number's relative position.

For inequalities like \(2 \leq x \leq 4\), the number line helps show which numbers satisfy this condition. You mark the specific numbers involved—here, 2 and 4—on the line. These numbers are boundaries for our specific set.

• Mark numbers in increasing order from left to right.
• Ensure equal spacing between numbers for clarity.
• Include critical numbers, in this case, 2 and 4, visibly on the line.

Here, you’ll place a filled circle at 2 and another at 4, showing that these endpoints are included in the set. You connect these points with a solid line, indicating all numbers between them, including 2 and 4, are part of the solution.

Interval notation is a concise way to express a set of numbers within a certain range. It uses brackets and commas to efficiently represent the set on a number line. For example, the interval \( [2, 4] \) denotes all numbers between 2 and 4, including the endpoints.

Brackets convey important information:

• Square brackets \([\ ]\) denote that the endpoint is included in the set.
• Round brackets \((\ )\) denote that the endpoint is not included.

For the set \( \{x \mid 2 \leq x \leq 4\} \), the interval notation \([2, 4]\) is used, meaning both 2 and 4 are part of the interval. This formalism helps quickly grasp which numbers belong to the set without drawing the number line.

Inequality symbols are used to show the relationship between two values, particularly if one value is larger or smaller than another. In the expression \(2 \leq x \leq 4\), two inequality symbols are used:

• \(\leq\): This symbol means "less than or equal to." The number is included in the set.
• <\(\geq\): Conversely means "greater than or equal to." This is used similarly to include numbers in a set.

These symbols are crucial for defining the range of values. Here, they tell us that \(x\) includes both 2 and 4, as well as every number in between. Understanding these symbols is vital for correctly interpreting and graphing inequalities on the number line.