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Interval notation is a mathematical way of representing a set of numbers along a continuous interval. It efficiently communicates which numbers are included or excluded in the solution set of an inequality.
For instance, in the inequality \(2 \leq x \leq 8\), we use interval notation to show the solution set as \([2, 8]\). Here, the square brackets \([ \text{and} ]\) mean that the endpoints 2 and 8 are included in the set.
This is because the inequality uses the \(\leq\) symbol, indicating that both 2 and 8 satisfy the inequality condition.
It’s important to note the difference between square brackets and parentheses \(( \text{and} ))\).

• If an endpoint is not included in the set, perhaps indicated by a \(<\) or \(>\) symbol in the inequality, we would use parentheses instead. For instance, \((2, 8]\) means values greater than 2 up to and including 8.

Learning to express solutions using interval notation is crucial for concisely conveying the results of inequalities, especially in more advanced math courses.

Number line graphing is a visual method of representing solutions to inequalities. It helps illustrate which parts of the number line make the inequality true.
To graph an inequality like \(|x-5| \leq 3\), we solve it and find the solution set, which we represent on the number line.
In our case, the solution set is \([2, 8]\). We start by drawing a simple horizontal line, marking out scaled points of interest. Here, points 2 and 8 are critical as they are the boundaries.

When graphing this:

• We use a solid dot at 2 and 8, illustrating these values are included in the solution (since it's \(\leq\), not just \(<\)).
• The section of the line between 2 and 8 is shaded, indicating all these values satisfy the inequality.

This graphical representation is powerful because it provides an immediate visual understanding of which numbers satisfy the inequality across a continuum rather than listing discrete values.

Compound inequalities involve more than one inequality statement joined by 'and' or 'or'. They allow us to find a range of solutions by solving multiple conditions simultaneously.
The inequality \(|x-5| \leq 3\) translates into the compound inequality \(-3 \leq x-5 \leq 3\), which naturally combines using 'and', implying both conditions must be true for the inequality to hold.
This can be broken into:

• \(-3 \leq x-5\)
• \(x-5 \leq 3\)

Each piece is solved separately.

• For \(-3 \leq x-5\), solving gives \(2 \leq x\).
• For \(x-5 \leq 3\), solving gives \(x \leq 8\).

Both solutions coexist, meaning \(2 \leq x \leq 8\), forming the complete solution.
This illustrates how compound inequalities work together to define a broader range of acceptable values in one cohesive set.