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An inequality is a mathematical statement that expresses the relative size or order of two values. It shows if one value is larger or smaller than another. There are four basic inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

In our specific exercise, the inequality \(8 \leq x \leq 10\) indicates that x can be any number from 8 to 10, *inclusive*. This means it includes the endpoints 8 and 10.

Inequalities are useful in various aspects of mathematics, including solving equations and graphing.

Key points to remember about inequalities include:
- The difference between strict (\(<\),\(>\)) and inclusive (\(\leq\),\(\geq\)) inequalities
- How to manipulate and solve inequalities to find the range of solutions
- How to represent inequalities on a number line

A number line is a straight line that visually represents numbers at equal intervals along its length. It provides a way to showcase numerical values, both positive and negative, in a linear format.

To use a number line for graphing inequalities or intervals, follow these steps:

• Identify and mark the key points required for your specific problem.
• Place solid or open dots on these points based on whether they are included in the interval (solid dot) or not included (open dot).
• Shade the segment between the points if they encompass a range of values.

Graphs on a number line help in understanding and visualizing the scope of the numbers you are dealing with.

In our example, \([8,10]\), both 8 and 10 are included (so we use solid dots), and we shade the line between these points to indicate all values within this interval.

Graphing intervals involves visualizing the range of values that satisfy a given inequality on a number line. This method helps in comprehending the solution set to an inequality.

To graph the interval \([8,10]\) on a number line:

• Draw a horizontal line to represent your number line.
• Mark the points 8 and 10 on the line.
• Place a solid dot (•) at both 8 and 10 because these endpoints are included in the interval (inclusive).
• Shade the line segment between 8 and 10 to represent all numbers in this range.

This shaded segment shows that any number from 8 to 10 can be a solution to the inequality \(8 \leq x \leq 10\).

Graphing intervals on a number line is a simple yet powerful tool for visualizing and solving inequalities. It gives a clear and intuitive understanding of which values are included in the solution set.