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A number line is a visual representation of numbers in order. Imagine it as a horizontal line where each point corresponds to a number. In mathematics, number lines help illustrate concepts like addition, subtraction, and, in our case, inequalities.

To draw a number line, start by marking a central point, often '0'. From there, place equal increments for positive and negative numbers on both sides. This way, you can graphically represent a range of values, such as those in an inequality. By marking specific numbers, like '7' in our problem, you gain a visual sense of the relationship between numbers.

When graphing inequalities, such as \(x \leq 7\), a number line helps see not just the specific number but what's included to the left or right. This handy tool makes abstract concepts much clearer.

A closed circle on a number line indicates that a number is included in the solution set of an inequality. It's like putting a dot over a number to say, "Yes, this number counts too!"

In the inequality \(x \leq 7\), placing a closed circle on '7' shows that 7 itself is a valid solution. If the inequality was strict, like \(x < 7\), we'd use an open circle instead. This small detail makes a big difference. It tells us if the boundary number is part of the solutions or not.

So, when you see a closed circle, remember: the number it's on should be considered included just like all the values to its left or right, depending on the inequality's direction.

The inequality \(x \leq 7\) uses the symbol '≤', meaning "less than or equal to". This tells us that \(x\) can be 7, or any number smaller. It's important to understand this concept to correctly graph and solve inequalities.

'Less than or equal to' combines two ideas:

• 'Less than': Values that are strictly smaller than the number (e.g., 6, 5, etc.).
• 'Equal to': Exactly the number itself (e.g., 7 in this case).

This understanding is key when graphing the inequality. The number line will be filled with a solid line or shading to the left of the number, along with a closed circle on the number itself. This visualization clearly represents the complete set of possible values for \(x\).