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To solve absolute value inequalities like \(|x| > 3\), it's important to remember what the absolute value represents. Absolute value measures the distance of a number from zero on the number line. When we say \(|x| > 3\), we are looking for all numbers whose distance from zero is more than 3.

The inequality can be broken into two separate cases because of the nature of absolute values. So, \(|x| > 3\) becomes:

• \(x > 3\)
• \(x < -3\)

This is because any number greater than 3 or less than -3 will have an absolute value greater than 3.
By solving these separate inequalities, we are determining the range of values for \(x\) that satisfy the original inequality.

So, the final solution to \(|x| > 3\) is that \(x\) can be any number greater than 3 or less than -3.

Graphing inequalities involves visually representing the solutions on a number line. For our inequality \(|x| > 3\), we have found two separate solution sets: \(x > 3\) and \(x < -3\). Each of these solutions needs to be graphed clearly to indicate which numbers satisfy the inequality.

For \(x > 3\):

• Place an open circle on 3 to indicate that 3 is not included in the solution.
• Shade the number line to the right of 3, showing all numbers greater than 3 are solutions.

For \(x < -3\):

• Place an open circle on -3 to indicate that -3 is not included in the solution.
• Shade the number line to the left of -3, indicating all numbers less than -3 are solutions.

This graph allows us to visually understand the areas on the number line that satisfy the inequality, showing two distinct regions.

Representing solutions on a number line is a key part of understanding inequalities. A number line provides a clear visual depiction of all possible solutions.

To construct the number line for \(|x| > 3\):

• First, draw a horizontal line and mark key points like -3 and 3.
• Place open circles at -3 and 3 to demonstrate that these specific numbers are not part of the solution.
• Shade the areas extending to the left of -3 and to the right of 3, signifying all the numbers in these regions are solutions.

The shaded areas show all numbers for which the original inequality holds true. This helps in understanding the continuous set of solutions and is an effective way for visual learners to see all the possibilities at a glance.