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When dealing with absolute value inequalities, a vital step involves expressing the problem as a compound inequality. This approach stems from the definition of absolute value, which measures how far a number is from zero without considering direction. For instance, if you have an inequality like \(|x| \leq 4\), it signifies that the distance of \(x\) from zero is at most 4. This can be translated into a compound inequality, consisting of two separate inequalities:

• \(-4 \leq x\)
• \(x \leq 4\)

These two inequalities capture the range of values \(x\) can take to satisfy the original absolute value condition. Similarly, in the problem \(|2x - 1| \leq 2\), transforming it into a compound inequality involves writing it as \(-2 \leq 2x - 1 \leq 2\).By solving each part separately, you can establish the range of solutions where the initial inequality holds true for \(x\). This method not only provides a clearer understanding of the solution domain but also ensures all potential solutions are considered.

Interval notation is a succinct way of expressing a range of values in inequalities. This notation is used to describe the set of numbers that satisfy a particular inequality. The primary advantage of interval notation is its compactness, making it easy to read and write.The interval notation uses brackets:

• Square brackets \([ \text{ or } ]\) indicate that the endpoint is included in the interval (closed interval).
• Parentheses \(( \text{ or } )\) suggest the endpoint is not included (open interval).

For example, in the solution \([-\frac{1}{2}, \frac{3}{2}]\), both \(-\frac{1}{2}\) and \(\frac{3}{2}\) are included in the set since square brackets are used.This interval notation stands for all values from \(-\frac{1}{2}\) up to \(\frac{3}{2}\), inclusive of the endpoints, indicating that the solution set includes every permissible \(x\) between these limits. Recognizing how to switch between inequalities and interval notation can strengthen your problem-solving toolkit, as it often helps to precisely communicate the solutions.

Graphing inequalities is a visual way to show the solutions of an inequality on a number line. This graphical representation is particularly useful in grasping the concept of range solutions.To graph the solution \([-\frac{1}{2}, \frac{3}{2}]\), you would:

• Draw a horizontal line representing the number line.
• Locate the points \(-\frac{1}{2}\) and \(\frac{3}{2}\) on this line.
• Shade the region between \(-\frac{1}{2}\) and \(\frac{3}{2}\) to indicate all the numbers in this range satisfy the inequality.
• Place a solid dot, or point, on \(-\frac{1}{2}\) and \(\frac{3}{2}\) to show that these endpoints are included as per the closed interval.

A shaded region conveys all the real numbers in this interval, providing a clear picture of the values that are solutions to the inequality. Graphing is not only about marking numbers but illustrating the concept that an inequality includes a continuum of values. By mastering this skill, you enhance your ability to visualize and interpret solutions in a meaningful way.