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The number line is an essential tool for understanding absolute value inequalities. It is a visual representation of numbers in a horizontal line where each point corresponds to a unique number. The center of this line, marked by 0, is crucial when working with absolute values. Absolute value measures the distance from a number to 0 on this line.

Imagine this exercise as searching for the points further away than 3 units from 0. With \(|x| > 3\), you need to find all points either more than 3 units to the right or the left on the number line. This means our solutions include any point greater than 3 and any point less than -3. Visualizing these on the number line helps confirm the solution. It’s a good practice to sketch these out, marking clear intervals that represent the solution set.

Inequalities are statements about the relative size or order of two numbers. In our exercise, the inequality \(|x| > 3\) signifies that we are comparing the distance of \(|x|\) from zero to 3. Understanding inequalities involves recognizing the symbols used:

• > : Greater than
• < : Less than

Here, \(|x| > 3\) implies two cases: \(x > 3\) or \(x < -3\). These cases arise because the absolute value function concerns distance, which can be approached from both sides of zero. When dealing with inequalities like these, it's vital to interpret them as instructions for dividing the number line into intervals that meet the condition.

Once you identify the intervals on the number line satisfying the inequality, you form the solution set. For \(|x| > 3\), the solution set includes all numbers beyond a distance of 3 units from zero. Thus, it is written as two separate intervals: \(x > 3\) and \(x < -3\).

These intervals align with the inequality's requirement that values of \(|x|\) must exceed 3. The solution set can be expressed as (-∞, -3) \cup (3, ∞). This union of intervals signifies that any number fitting either condition is part of the solution. This form of representation helps us understand which numbers satisfy the original inequality by listing all possible solutions.