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The absolute value of a number is its distance from zero on the number line. It's always a nonnegative number. Whether the number is positive or negative, you take off the sign and keep the magnitude. For example, the absolute value of both 5 and -5 is 5.

In inequalities involving absolute values, like \(|x| > 3\), we're interested in finding numbers whose distance from zero exceeds 3. This can result in two situations:

• Numbers greater than 3
• Numbers less than -3

This is because anything less than -3 or more than 3 fulfills the condition of being over 3 units away from zero. Therefore, absolute values are a powerful way to express distances on the number line.

When you encounter an absolute value inequality, it's typically split into two simpler inequalities to solve each separately.

The number line is a visual tool that represents all real numbers as points on a straight line. It's like a ruler but extends infinitely in both directions, with zero in the middle.

Using a number line to solve inequalities helps us visually understand where solutions fit. For example, when solving \(|x| > 3\), you can imagine all points greater than 3 and all points less than -3 marked on this line.

This effective method of representation makes it easier to comprehend the solution sets visually. A number line helps highlight the open intervals where solutions exist, reflecting inequalities ranging across multiple intervals.

Graphing inequalities on a number line visually represents all possible solutions to the inequality. When graphing \[|x| > 3\], we create a straightforward picture of the intervals that satisfy the inequality. Here's a simple way to do it:

• Draw the number line.
• Locate the point 3 and -3.
• Use an open circle on both points to indicate the value itself is not included (since inequality signs are 'greater than', not 'greater than or equal to').
• Shade the line to the right of 3, and the line to the left of -3, to show all numbers greater than 3 and less than -3.

This graphing process clearly demonstrates the solution set of the inequality on the number line, confirming that any number greater than 3 or less than -3 is part of the solution.