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Imagine a long, horizontal line that continues infinitely in both directions. This is what we call a number line. It's a tool that helps us visualize the position of numbers in relation to each other. On the number line, every point corresponds to a real number. To use this tool for graphing inequalities, we start by identifying and marking reference points - typically integers - and scaling the line consistently.
For inequality solutions, we place marks to represent the boundary numbers of the inequality. For example, if we consider the inequality \(x \leq 3\), the number 3 becomes our reference point. The act of graphing on a number line makes abstract concepts more concrete, helping us to understand and interpret inequalities effectively. By marking the exact point where 3 lies on the line and indicating the direction that satisfies the inequality, we can visually summarize the range of possible solutions.

The symbols used in inequalities deliver crucial information about the relationship between values. Common inequality symbols include \(<\), \(>\), \(\leq\), and \(\geq\). The symbol \(<\) means 'less than', and \(>\) means 'greater than'. When these are accompanied by the equal sign as in \(\leq\) or \(\geq\), that means they are 'less than or equal to' and 'greater than or equal to', respectively. These symbols dictate the direction we shade or draw on the number line.
In our exercise, \(x \leq 3\) uses the 'less than or equal to' symbol, indicating that all numbers less than 3, plus 3 itself, satisfy the inequality. Understanding these symbols is key to interpreting the solution correctly and to applying the right graphical representation on the number line. For instance, the way we draw a line or a ray on the number line depends on whether the inequality includes the boundary number (closed circle notation) or not (open circle notation).

In graphing inequalities, closed circle notation plays a significant role. It's the visual cue that the number where the circle is drawn is included in the set of solutions. We use a solid dot, or 'closed circle', right at the number that is part of the solution. For instance, the inequality \(x \leq 3\) means that 3 is a valid solution to the inequality, in addition to all numbers less than 3. So, we mark the 3 on our number line with a closed circle.
Additionally, this closed circle connects to a line or ray that extends in the direction of the other numbers which also satisfy the inequality. In the provided exercise, after placing the closed circle on 3, we draw a line going left to show all numbers less than 3 are also included. Remember that the opposite of this notation is the 'open circle', which we would use if the boundary number were not included, such as with \(x < 3\). The choice between open and closed circle notation is crucial for conveying accurate information about the inequality's solutions.