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When we talk about graphing inequalities, like the inequality \(x \leq 4\), we are using a visual way to display the range of numbers that satisfy this inequality. In this case, it means showing all the numbers that are less than or equal to 4. Orders such as "less than" or "greater than" occur frequently when working with inequalities. To graph this, we first start with a number line, ensuring it spans all relevant values, particularly highlighting the point 4.

The symbols and tools we use include:

• An arrow or line to indicate the continuous nature of the numbers that fit the inequality.
• A marker or circle on the number line that helps us highlight the key boundary number, like 4 in this case.

Once these are plotted, our graph visually showcases every number that makes the inequality true. This method makes it easier for us to understand the set of solutions at a glance.

Real numbers encompass all the numbers we commonly use, including whole numbers, fractions, and decimals - essentially any number that can appear on the number line. They can be positive, negative, or zero.

When dealing with inequalities involving real numbers, we often think about how these numbers can stretch infinitely in both the positive and negative directions. In the context of our graph, every single number that lies on the left side of 4, as well as 4 itself, defines the real number set for \(x \leq 4\). This range is infinite, meaning it does not stop at any point along the left side, extending to include negative numbers as well.

Understanding real numbers helps us when choosing which areas of the number line to shade or mark to ensure the correct set of numbers is represented.

In the world of graphing, a closed circle is a simple yet powerful symbol. It reflects inclusion of the endpoint or boundary number, like the number 4 in our example. Whenever you see a closed circle on a number line, it signifies that the number beneath it is part of the solution set.

Here's why we use a closed circle:

• It clearly denotes that the boundary is included in the set of solutions.
• For the inequality \(x \leq 4\), 4 itself is valid and should be included, necessitating a closed circle.

Unlike an open circle, which would represent numbers strictly less than or greater than a certain number but not including it, the closed circle is integral in visualizing inequalities where the boundary number is indeed part of the solution set. This distinction ensures accuracy and clarity in your graphs.