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Understanding how to graph inequalities is a fundamental skill in algebra that allows students to visualize the range of solutions to an inequality. In the given exercise, we're presented with a compound inequality, which means it involves two distinct inequalities linked together, specifically, \(3 < x-2 < 7\).

After adding 2 across the inequality, we find that \(5 < x < 9\). To graph this, imagine a horizontal line with numbers marked at regular intervals. You'll plot two points, one at 5 and another at 9. Since 5 and 9 are not part of the solution, we illustrate this by drawing hollow circles at these points. Solid circles would indicate that those numbers are included in the solution.

The next step is to connect these hollow circles with a line, which represents all the numbers between 5 and 9. This line on the number line effectively captures all the possible solutions for the variable \(x\).

The number line is a visual representation of numbers laid out in a straight path, allowing students to easily understand numerical relationships and solutions. When working with inequalities, it is important to correctly use the number line as a tool to represent the range of solutions. A number line extends infinitely in two directions, positively to the right and negatively to the left.

For our inequality \(5 < x < 9\), the number line must clearly show that x can be any real number greater than 5 and less than 9. The hollow circles are crucial as they convey exclusion, indicating that 5 and 9 are limits that x cannot equal. When using a number line to show intervals, always ensure the correct use of hollow or solid circles to reflect the problem accurately.

Interval notation is a concise way of writing the set of all possible solutions to an inequality. It is a neat and universal method used to describe intervals on a number line. In interval notation, parentheses, \(()\), are used to denote that an endpoint is not included in the interval, while brackets, \([]\), signify inclusion.

For our problem's solution \(5 < x < 9\), we use parentheses to signify that neither 5 nor 9 are included in the solution set. This would be written in interval notation as \((5, 9)\). This notation is essentially a condensed version of the number line graph we created, capturing the idea that x can take any real value between, but not including, 5 and 9.

When reading or writing in interval notation, always pay close attention to the type of bracket or parenthesis. It provides essential information about the limits of the interval.