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Interval notation is a way of representing a range of numbers along the number line that form the solution to an inequality. It is a concise and standardized method to denote intervals. In our solution to the inequality \(|x-5| \leq 3\), we found that the solution set is all numbers between 2 and 8, inclusive. Hence, the interval notation is written as \([2, 8]\). Here’s how to interpret it:

- **Brackets** "[ ]" indicate that the endpoints are included in the solution. These are known as "closed intervals." In our example, 2 and 8 are part of the solution, so we use "[2, 8]".
- **Parentheses** "( )" would be used if the endpoints were not included, known as "open intervals." For instance, "(2, 8)" would include numbers greater than 2 and less than 8, but not including 2 and 8 themselves.

This method effectively communicates the range of possible values for \(x\) in a neat and compact form.

The number line graph is a visual representation of the solutions to an inequality. It helps to clearly see the range of possible values. For our inequality solution \(|x-5| \leq 3\), the number line graph features a part of the line that is shaded between 2 and 8.

Here's how you set up this graph:

• Draw a horizontal line to represent the number line.
• Mark points on this line at 2 and 8.
• Use solid dots at both 2 and 8 to indicate that these endpoints are included in the solution.
• Shade the line between 2 and 8.

This graphically shows that all values from 2 to 8 are included in the solution set. It provides a quick visual cue to understand the range of solutions.

Solving absolute value inequalities involves understanding that the expression within the absolute value can be within a range both positive and negative. For \(|x-5| \leq 3\), solving leads to the range of numbers that satisfy the original inequality.

Steps to find the solution:
- **Set Up Two Inequalities:** The absolute value inequality \(|x-5| \leq 3\) can be split into two separate inequalities:
1. \(-3 \leq x-5\)
2. \(x-5 \leq 3\)

- **Solve Each Inequality:**
1. From \(x - 5 \leq 3\), add 5 to both sides, resulting in \(x \leq 8\).
2. From \(x - 5 \geq -3\), add 5 to both sides, bringing us to \(x \geq 2\).

- **Combine Solutions:** Together these give \(2 \leq x \leq 8\), meaning \(x\) can be any number from 2 to 8 inclusive.

The solution to this inequality is a range expressed both in interval notation \([2, 8]\) and visually represented on a number line graph. It shows all the values that \(x\) can take while satisfying the given inequality.