91Ó°ÊÓ

Solving inequalities involves finding all possible values of a variable that satisfy a particular condition. For the inequality \(|x| > 3\), we have to determine the values of \(x\) where the absolute value of \(x\), which is the distance from zero, is greater than 3. Because absolute inequalities often involve separate conditions on both sides of the number line, interpreting them correctly is key.
To solve \(|x| > 3\), we break it down into two inequalities:

• \(x > 3\) — The values of \(x\) that are more than 3 units away on the positive side.
• \(x < -3\) — The values of \(x\) that are more than 3 units away on the negative side.

By splitting it into these simpler inequalities, we can handle each part independently. This allows us to find the complete range of solutions that satisfy the original inequality. Each inequality represents a straightforward condition to solve, and we combine these solutions in the final step needed for graphing.

Representing inequalities on a number line is a visual way of showing which values satisfy the inequality. For the inequalities \(x > 3\) and \(x < -3\), our solution doesn’t include the points \(3\) and \(-3\) themselves. Instead, we represent them with open dots, indicating that these numbers are not part of the solution.
To graph these:

• Place an open dot at \(3\), and draw an arrow extending to the right to show that all numbers greater than \(3\) are part of the solution.
• Place an open dot at \(-3\), and draw an arrow extending to the left to show that all numbers less than \(-3\) are part of the solution.

This approach effectively visualizes the two separate ranges, which are not connected but both satisfy the original inequality. The number line graph makes it easier to understand the extent and partition of solutions.

Compound inequalities involve two or more separate inequalities linked by the words 'and' or 'or'. In the case of \(|x| > 3\), we have an 'or' situation, meaning we are looking for values of \(x\) that satisfy either inequality: \(x > 3\) or \(x < -3\).
The 'or' represents that \(x\) needs fulfilling either one of the conditions to be part of the solution. This results in two distinct intervals on the number line. Unlike 'and' compound inequalities, 'or' allows for solutions in multiple pieces.
For example:

• \(x > 3\) means all numbers to the right of 3.
• \(x < -3\) means all numbers to the left of -3.

Together, these intervals capture the complete set of solutions, highlighting how absolute inequalities create compound scenarios. Understanding how to address both conditions allows you to thoroughly solve a broader range of complex inequalities.