Problem 56 - Precalculus

Question

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-1| \leq \frac{1}{2}$$

Step-by-Step Solution

Answer

x is in the interval [1/2, 3/2].

Step 1: Understanding the Inequality

The inequality given is \(|x-1| \leq \frac{1}{2}\). This means that the absolute value of the expression \(x-1\) needs to be less than or equal to \(\frac{1}{2}\). We need to find values of \(x\) that satisfy this condition.

Step 2: Convert Absolute Inequality into Compound Inequality

To remove the absolute value, we split the inequality \(|x-1| \leq \frac{1}{2}\) into two separate inequalities: \(x - 1 \leq \frac{1}{2}\) and \(-(x - 1) \leq \frac{1}{2}\). This accounts for both the positive and negative scenarios around the absolute value.

Step 3: Solve the First Inequality

We solve the inequality \(x - 1 \leq \frac{1}{2}\). To do this, we add 1 to both sides to isolate \(x\):\[x \leq \frac{1}{2} + 1 = \frac{3}{2}.\]

Step 4: Solve the Second Inequality

We solve the second inequality \(-x + 1 \leq \frac{1}{2}\). First, subtract 1 from both sides to get:\[-x \leq \frac{1}{2} - 1 = -\frac{1}{2}.\]Then, multiply both sides by -1, which reverses the inequality to:\[x \geq \frac{1}{2}.\]

Step 5: Combine the Solutions

The solutions \(x \leq \frac{3}{2}\) and \(x \geq \frac{1}{2}\) mean that \(x\) lies between \(\frac{1}{2}\) and \(\frac{3}{2}\). Thus, we write the final solution as an interval:\[\frac{1}{2} \leq x \leq \frac{3}{2}.\]

Step 6: Represent the Interval on the Number Line

On the number line, this solution is represented by a solid line segment between \(\frac{1}{2}\) and \(\frac{3}{2}\), including both endpoints since the inequality is non-strict. This indicates that all values of \(x\) within and including \(\frac{1}{2}\) and \(\frac{3}{2}\) are solutions.

Key Concepts

Compound Inequalities

Compound inequalities involve understanding and solving inequalities that are connected by conjunctions such as "and" or "or." In the case of absolute value inequalities like \(|x-1| \leq \frac{1}{2}\), we use compound inequalities to express the solution set. This involves converting the expression into two distinct linear inequalities.
The key is to split the absolute value inequality into two cases to account for the scenarios where the value inside the absolute sign could be either positive or negative. For our example, we have:

• \(x - 1 \leq \frac{1}{2}\) – handling the positive case.
• \(-(x - 1) \leq \frac{1}{2}\) – accounting for the negative aspect.

Solving each of these inequalities separately gives us a range of values that satisfy the condition, leading to a compound inequality that describes all possible solutions between two limits. Compound inequalities help narrow down the specific parts of the number line where solutions can be found, making more sense of seemingly complex problems.

Number Line Representation

Number line representation provides a visual depiction of solutions that makes understanding inequalities more intuitive. Once we have established the range of values for \(x\) using compound inequalities, we can place these on a number line to clearly see where the solutions lie.
For the inequality \(\frac{1}{2} \leq x \leq \frac{3}{2}\), the number line serves as a simple way to illustrate these solutions. This interval is represented as:

• A solid line between \(\frac{1}{2}\) and \(\frac{3}{2}\), indicating continuous solutions.
• Closed circles or brackets at \(\frac{1}{2}\) and \(\frac{3}{2}\), representing the inclusion of these endpoints.

By marking these intervals on the number line, it’s easier to visualize which values of \(x\) satisfy the original inequality. This method is invaluable for quickly showing solutions, especially when dealing with problems that require a visual component for clarity.

Interval Notation

Interval notation is a concise and mathematical way of expressing the solution set of inequalities. It's especially useful for communicating compound inequalities after solving absolute value problems. In the exercise given, we've found that \(x\) is between \(\frac{1}{2}\) and \(\frac{3}{2}\).
This is expressed in interval notation as \([\frac{1}{2}, \frac{3}{2}]\). The brackets signify that both endpoints \(\frac{1}{2}\) and \(\frac{3}{2}\) are included in the solution. If an endpoint were not included, we’d use a parenthesis instead of a bracket.
One key aspect of interval notation is its ability to simplify the representation of solution sets. By using just a couple of symbols and numbers, interval notation encapsulates the entire solution compactly, making it ideal for use in both written and digital formats. Understanding interval notation is essential for effectively communicating solutions across various mathematical contexts.