Problem 67 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ \left|2 x-\frac{1}{3}\right|>\frac{2}{3} $$

Step-by-Step Solution

Answer

\( x < -\frac{1}{6} \) or \( x > \frac{1}{2} \)

Step 1: Understand Absolute Inequality Form

The inequality is of the form \( |A| > B \), which means that the solution is when \( A > B \) or \( A < -B \). This form helps us break down the inequality into two separate inequalities to solve.

Step 2: Break Down into Two Inequalities

Using the absolute inequality rule, we break down \( |2x - \frac{1}{3}| > \frac{2}{3} \) into two separate inequalities:1. \( 2x - \frac{1}{3} > \frac{2}{3} \)2. \( 2x - \frac{1}{3} < -\frac{2}{3} \)

Step 3: Solve First Inequality

For \( 2x - \frac{1}{3} > \frac{2}{3} \), start by adding \( \frac{1}{3} \) to both sides:\[ 2x > \frac{2}{3} + \frac{1}{3} \]This simplifies to:\[ 2x > 1 \]Now, divide both sides by 2:\[ x > \frac{1}{2} \]

Step 4: Solve Second Inequality

For \( 2x - \frac{1}{3} < -\frac{2}{3} \), start by adding \( \frac{1}{3} \) to both sides:\[ 2x < -\frac{2}{3} + \frac{1}{3} \]This simplifies to:\[ 2x < -\frac{1}{3} \]Now, divide both sides by 2:\[ x < -\frac{1}{6} \]

Step 5: Combine Solutions

The solution to the inequality \( |2x - \frac{1}{3}| > \frac{2}{3} \) are the values of \( x \) that satisfy either \( x > \frac{1}{2} \) or \( x < -\frac{1}{6} \). Therefore, the combined solution is:\[ x < -\frac{1}{6} \quad \text{or} \quad x > \frac{1}{2} \]

Key Concepts

Understanding Absolute Value

In mathematics, the concept of absolute value is crucial for understanding how distance works on a number line. Absolute value measures how far a number is from zero, without considering which direction it is. It is always non-negative. For example, both 3 and -3 have an absolute value of 3, represented as \(|3| = 3\) and \(|-3| = 3\). This concept helps us interpret various algebraic problems and graphs.

Absolute value expressions are commonly found in real-world applications like calculating distance or error margins. It's like only caring about how far you are from your destination, not the direction you're heading. Thus, absolute value is represented with two vertical bars: \(|A|\). If \(|A| = B\), then the essence is that the distance A is at B distance from the origin, zero. This sets the foundation for equations and inequalities involving absolute values.

Inequality Solutions

Solving inequalities is similar to solving equations but with more nuances. In an inequality, you find a range of values rather than a single solution. For instance, consider an inequality to determine when one expression becomes greater than another.

For the inequality \(|A| > B\), \(A\) could be larger than \(B\) or less than \(-B\). This creates two potential situations to evaluate, leading to the understanding that we need to find two separate ranges for \(A\), or in algebraic terms, transform this into separate inequalities: \(A > B\) and \(A < -B\).

This methodology helps us better understand and visualize the solution set on a number line. The solution to an absolute inequality is often expressed in interval notation, which precisely defines the possible values that satisfy the inequality. Thus, inequalities help broaden the comprehension of the relationships between algebraic expressions.

Solving Inequalities Step by Step

Solving inequalities systematically can help avoid confusion and errors. The problem at hand was \(|2x - \frac{1}{3}| > \frac{2}{3}\). Let's dissect this step by step, using the approach outlined previously, to determine the solution.

First, we recognize the form \(|A| > B\), depicting two scenarios: \(A > B\) or \(A < -B\). Break it down:

• For \(2x - \frac{1}{3} > \frac{2}{3}\): Add \(\frac{1}{3}\) to both sides to simplify it to \(2x > 1\). Then divide by 2 to find \(x > \frac{1}{2}\).


• For \(2x - \frac{1}{3} < -\frac{2}{3}\): Similarly, add \(\frac{1}{3}\) and simplify to \(2x < -\frac{1}{3}\). Divide by 2 to solve for \(x < -\frac{1}{6}\).

Finally, interpret these results. The solution is a union of the two intervals: \(x < -\frac{1}{6}\) or \(x > \frac{1}{2}\). It means any value of \(x\) that fits into these ranges satisfies the original inequality, visually forming two separate intervals when plotted. Armed with this structured approach, tackling similar inequality problems becomes much more approachable.