Problem 28 - Calculus: One and Several Variables

Question

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

Step-by-Step Solution

Answer

The solution to the inequality is \(-\frac{1}{2} < x < 0 ∪ 0 < x < \frac{1}{2}\) or in interval notation: \((-\frac{1}{2}, 0) ∪ (0, \frac{1}{2})\).

Step 1: Set up the inequalities

Because of the absolute value, the expression \(|x|\) can either be a positive or negative \(x\). Therefore, a representation of \(0<|x| < \frac{1}{2}\) can be written as \(0< x < \frac{1}{2}\) and \(-\frac{1}{2}< x < 0\).

Step 2: Solve the inequalities

In this step, there is nothing to solve because the x-variable is already isolated and it is clear that for the first inequality, \(x\) lies between 0 and \(\frac{1}{2}\), and for the second inequality, \(x\) lies between -\(\frac{1}{2}\) and 0.

Step 3: Expressing the solution as an interval

The union of the solution sets from step 2 gives the complete set of solutions for the inequality. This can be written as an interval like this: \(-\frac{1}{2} < x < 0\) or \(0 < x < \frac{1}{2}\). Using the union symbol '∪' to combine these gives \(-\frac{1}{2} < x < 0 ∪ 0 < x < \frac{1}{2}\). In interval notation, the solution may be represented as \((-\frac{1}{2}, 0) ∪ (0, \frac{1}{2})\).

Key Concepts

Absolute Value Inequalities

Understanding absolute value inequalities is crucial for solving problems that involve distances and measurements in algebra. An absolute value represents the distance of a number from zero on a number line, regardless of direction, making it always nonnegative. Inequalities involving absolute values, such as \(0<|x| < \frac{1}{2}\), inform us that the variable enclosed can take on values within a certain range from zero.

When tackling an absolute value inequality, visualize splitting the inequality into two separate scenarios: one where the variable is positive, and another where it's negative. For the given example, we have two inequalities, \(0< x < \frac{1}{2}\) when \(x\) is positive, and \(-\frac{1}{2}< x < 0\) when \(x\) is considered negative. By solving these, we determine all possible values of \(x\) that satisfy the original absolute value inequality. This approach simplifies the process, allowing us to easily capture the full range of solutions.

Interval Notation

Interval notation is a streamlined method to express the set of all numbers between two endpoints. In essence, it is a notational shorthand that uses parentheses and brackets to describe ranges on the number line. For open intervals, where an endpoint is not included in the solution set, parentheses are used. In contrast, closed intervals that include the endpoints are denoted with brackets.

Consider our example, which results in two open intervals: \((-\frac{1}{2}, 0)\) and \((0, \frac{1}{2})\). Why parentheses? Because both the 0 and \(-\frac{1}{2}\) are not part of the solutions to the inequality; \(x\) is strictly less than \(\frac{1}{2}\) and strictly greater than \(-\frac{1}{2}\). Interval notation enables a concise and clear presentation of these solution sets and is often preferred when graphing inequalities or communicating mathematical ideas efficiently.

Inequality Solutions

Navigating the solutions to inequalities, especially those with absolute values, demands attention to the range of permissible values for the variable. The final step in solving an inequality like \(0<|x| < \frac{1}{2}\) is to combine the separate ranges into a comprehensive solution. This involves using a mathematical union (denoted by '∪'), which essentially means 'and/or'.

In our scenario, we have two ranges for \(x\), captured by the individual inequalities \(-\frac{1}{2} < x < 0\) and \(0 < x < \frac{1}{2}\). The combined solution is then clearly stated as \(-\frac{1}{2} < x < 0 ∪ 0 < x < \frac{1}{2}\), which includes all the values that \(x\) could be to satisfy the original inequality. The solution set is a union of two open intervals, demonstrating that there is a whole range of values – though not contiguous – that answer the problem effectively. Inequality solutions thus provide insight into the possible values variables can take within the realm of mathematical conditions laid out by the inequality.