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A number line is a visual representation of numbers in a straight line. Each point on the line corresponds to a number. Number lines typically include both positive and negative numbers, often with zero at the center.
It functions as a helpful tool for visualizing various mathematical concepts, such as inequalities or arithmetic operations.
A number line is usually horizontal and labeled with numbers in equal intervals. This structure helps in comparing values or understanding distance between numbers.

• Negative numbers are placed to the left of zero.
• Positive numbers are placed to the right.
• The numbers increase as you move to the right, and decrease as you move to the left.

In the context of graphing inequalities, a number line allows us to represent all possible solutions of an inequality. You plot these solutions by using circles and shading areas on the line.

Inequality symbols are mathematical tools used to compare two numbers or expressions. They tell us the relationship between the values, indicating whether one is greater than, less than, or equal to another.

• "<" means less than.
• ">" means greater than.
• "≤" means less than or equal to.
• "≥" means greater than or equal to.

In the inequality \(-5 \geq x\), the symbol \(\geq\) signifies that \(x\) includes values that are less than or equal to \(-5\). It combines the concepts of both the "greater than" and "equal to" symbols. This is key to understanding how to graph the inequality as it indicates what values of \(x\) should be shown on a number line.

A closed circle on a number line indicates a specific value that satisfies an inequality. When you see a closed circle over a number, it means that this number is included in the solutions set.
This is different from an open circle, which means that the number is not included.
In our example, the closed circle is at \(-5\), reflecting the inequality \(-5 \geq x\). Because the symbol \(\geq\) includes \'equal to\', the closed circle ensures that \(-5\) is part of our answer and solution set.

Shading on a number line visually represents the range of possible values that satisfy an inequality. Once you have established where to place a circle (either open or closed), the next step is to shade the region that includes all valid solutions.
For the inequality \(-5 \geq x\), you start the shading at the closed circle placed at \(-5\).
Then, shade all points to the left of \(-5\) to represent numbers less than \(-5\).

• The shading extends indefinitely to the left.
• The shaded area communicates that all these numbers satisfy the inequality.
• An arrow at the end of the shaded area shows the continuation of the set of solutions beyond the shown interval.

This shading method makes it easy to see at a glance which numbers are part of the solution set and helps solidify understanding of numeric relationships in inequalities.