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Graphing inequalities, like \(|x| < 3\), involves representing a range of values on a graph, particularly on a number line. To start, you need to understand the inequality. In this case, \(|x| < 3\) means we are interested in all numbers whose absolute value, or distance from zero, is less than 3.

Once you rewrite the absolute value inequality as \(-3 < x < 3\), it becomes clear what numbers are part of the solution set. These are all the numbers between -3 and 3. They don't include -3 and 3 themselves, but any point between these two is a valid solution.

On a number line, you'll represent this by using open circles at -3 and 3. This choice indicates that the boundary numbers are not included in the solution. The segment on the number line between these open circles is then shaded or marked to show all the numbers that satisfy this condition.

Number lines are a crucial tool for visualizing inequalities like \(|x| < 3\). Imagine a straight line that represents all possible values of 'x'. Each point corresponds to a number, with negative numbers to the left and positive numbers to the right. Zero sits at the center.

When dealing with an inequality such as \(-3 < x < 3\), you focus on the segment of the number line between -3 and 3. This portion needs to be highlighted because the solutions to the inequality lie in this region.

To draw this on a number line:

• Place an open circle at -3, indicating that -3 is not included.
• Place another open circle at 3, for the same reason.
• Shade the line between these two circles to represent all numbers between them.

Number lines make it easy to see where the values that meet the inequality's condition are located.

Finding solutions to inequalities involves determining which values satisfy the given condition. In the exercise \(|x| < 3\), solving it means identifying numbers that have an absolute value less than 3.

Start by breaking down the absolute value inequality to regular inequalities: \(-3 < x < 3\). This step tells you directly which numbers 'x' can be. Every number within this range is a solution.

We exclude the boundaries -3 and 3 because the inequality is strict ("less than"), not "less than or equal to." Solutions are values that lie within these bounds.

Graph visually shows these solutions as the shaded interval between -3 and 3 with open circles at each end, confirming that the endpoints themselves aren't part of the solution set.

Inequality solutions help in understanding ranges and sets of numbers and are key in various mathematical applications and real-world scenarios.