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An inequality is a mathematical expression that shows the relationship between two values when they are not equal. In other words, it indicates whether one value is greater than, less than, greater than or equal to, or less than or equal to another.

Inequalities can come in various forms, such as the simple ones like \(x > 1\) or compound ones like \(|x| > 1\). When dealing with absolute values, the inequality \(|x| > 1\) tells us that we are interested in the numbers whose absolute value is greater than 1.

To solve such inequalities, think of them as two separate cases: one for the positive direction and one for the negative. For our example, it means solving \(x > 1\) and \(x < -1\) separately. This method ensures we cover all numbers whose absolute value exceeds 1, either positively or negatively.

The solution set is the collection of all possible values that satisfy a given equation or inequality. For the inequality \(|x| > 1\), the solution set comprises two distinct parts based on our split from the inequality analysis:

• Values where \(x > 1\), which include all numbers greater than 1.
• Values where \(x < -1\), which include all numbers less than -1.

This solution set can be represented using interval notation as \((-\infty, -1) \cup (1, \infty)\). This notation shows both sets of numbers separately and excludes the values -1 and 1 from the solution set due to the nature of the strict inequality ">". This is why the intervals are open, using parentheses instead of square brackets, which would indicate inclusion.

Understanding how to determine solution sets visually or through interval notation is key to grasping how inequalities work, especially when involving absolute values.

A number line is a simple visual tool that helps represent numbers and their relative values and positions. It's quite useful for showing the solution set of an inequality like \(|x| > 1\). When graphing this solution:

Start by drawing a horizontal line and marking critical points like -1 and 1. Since our solution set does not include these points (\(-1\) and \(1\) themselves), place open circles on -1 and 1 to indicate they are not part of the solution.

Then, shade everything to the left of -1 and everything to the right of 1, representing numbers that satisfy \(x < -1\) and \(x > 1\). As a result, the shaded parts indicate solutions where the absolute value is greater than 1, effectively graphing the solution for the inequality.