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The number line is a simple yet powerful visual tool in mathematics. It helps us to understand the ordering and relative size of numbers. Imagine it as a horizontal straight line that extends infinitely in both directions. To graph inequalities, like the one given in the exercise, a number line allows us to clearly see which numbers are included in the solution set.

When dealing with the inequality \( x \geq -10 \), the number line will show us where \(-10\) is, and which direction includes the greater numbers. Since \( x \) is greater than or equal to \(-10\), we focus on the right side of the number line. A solid dot on the number line at \(-10\) tells us that \(-10\) is part of the solution; thus, including all numbers larger than \(-10\). Using a shaded line extending to the right from this dot effectively communicates the range of solutions.

Graphing inequalities is an important technique to visually represent the solutions of inequalities. When you graph an inequality like \( x \geq -10 \), you use the number line to help you.

Here's a step-by-step process:

• Identify the boundary of the inequality. In this case, it is \(-10\).
• Since the inequality symbol \( \geq \) includes equality, you place a solid dot at \(-10\) to include it in the solution.
• Shade the number line to the right of \(-10\), since \( x \) is greater than or equal to \(-10\). This shaded portion shows all the possible solutions, extending infinitely to the right.

This graph visually confirms that any number greater than or equal to \(-10\) satisfies the inequality. It's a very intuitive way to check your solution.

Interval notation is another way to express the solution set of an inequality, using brackets to denote endpoints. It provides a compact expression of a range of numbers.

For the inequality \( x \geq -10 \), the endpoint \(-10\) is included, so you use a square bracket \([\) to show that \(-10\) is part of the solution. Since the numbers extend infinitely in the positive direction, you use the symbol \( \infty \) to indicate infinity, which is always accompanied by a parenthesis \()\) because infinity is never a reachable endpoint.

Thus, the interval notation for \( x \geq -10 \) is \([-10, \, \infty)\). This notation is a versatile mathematical shorthand that allows you to describe extensive ranges succinctly, handy in multiple mathematical contexts.