Problem 30 - Calculus: One and Several Variables

Question

Solve the inequality and express the solution set as an interval or as the union of intervals. $$0 < \left|x-\frac{1}{2}\right| < 2$$.

Step-by-Step Solution

Answer

The solution to the inequality \(0 < \left|x-\frac{1}{2}\right| < 2\) is the interval \(\left(\frac{1}{2}, \frac{5}{2}\right)\).

Step 1: Interpret the Absolute Value Inequality

Interpret the given absolute value inequality as two separate inequalities: \(0 < x - \frac{1}{2}\) and \(x - \frac{1}{2} < 2\). These inequalities represent the two cases that occur when the absolute value is removed.

Step 2: Solve the First Inequality

To solve \(0 < x - \frac{1}{2}\), add \(\frac{1}{2}\) to each side of the inequality to isolate \(x\). This results in \(x > \frac{1}{2}\).

Step 3: Solve the Second Inequality

Similarly, to solve the inequality \(x - \frac{1}{2} < 2\), add \(\frac{1}{2}\) to each side to isolate \(x\), giving \(x < \frac{5}{2}\).

Step 4: Combine the Inequalities

We have \(x > \frac{1}{2}\) from the first inequality and \(x < \frac{5}{2}\) from the second inequality. Therefore, the solution of the given absolute value inequality is the interval \(\left(\frac{1}{2}, \frac{5}{2}\right)\), which includes all 'x' values that satisfy both inequalities.

Key Concepts

Interval Notation

When we talk about interval notation, we are discussing a way to describe a range of numbers along the number line. It's a simplified way to show which numbers are included in a set. For example, to express all the values between \( \frac{1}{2} \) and \( \frac{5}{2} \), we write it as \((\frac{1}{2}, \frac{5}{2})\). In this notation:

• Parentheses \(()\) indicate that the endpoints are not included (open interval).
• Brackets \([]\) suggest that the endpoints are part of the set (closed interval).

So, \((\frac{1}{2}, \frac{5}{2})\) means every number greater than \(\frac{1}{2}\) and less than \(\frac{5}{2}\), but not the endpoints themselves. It’s an efficient way to convey the solution set of an inequality.

Inequality Solutions

Solving inequalities involves determining the range of values that satisfy a given condition. When confronted with an absolute value inequality like \(0 < |x - \frac{1}{2}| < 2\), it splits into two parts:

• \(0 < x - \frac{1}{2}\)
• \(x - \frac{1}{2} < 2\)

To solve each part, you break them down:

• Isolate \(x\) in \(0 < x - \frac{1}{2}\) to find \(x > \frac{1}{2}\).
• In \(x - \frac{1}{2} < 2\), solving gives \(x < \frac{5}{2}\).

By combining these, you determine the set of all \(x\) satisfying both conditions, expressed in interval notation as \((\frac{1}{2}, \frac{5}{2})\). This process reflects the logical structure needed to solve such inequalities.

Mathematical Problem Solving

When tackling mathematical problems, especially inequalities, understanding is key. Begin by interpreting the problem's statement accurately. With inequalities involving absolute values, break them into manageable parts. This approach helps in clarifying the range of possible solutions.

For the inequality \(0 < |x - \frac{1}{2}| < 2\), divide it into two separate inequalities. Solving these individually simplifies the process. Each step requires manipulating the equation by adding or subtracting terms or dividing by coefficients to isolate \(x\).

Combining the solutions from these simplified inequalities allows you to construct a larger solution set. Writing the answer in interval notation clearly communicates the solution, emphasizing both the calculated results and understanding of the underlying concepts. Always cross-check your answers to ensure that they logically fit within the parameters set by the original problem.