Problem 61 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$\left|6-\frac{1}{3} x\right|>0$$

Step-by-Step Solution

Answer

The solution is all real numbers except \( x = 18 \).

Step 1: Understand the Absolute Value Inequality

The expression \( |6 - \frac{1}{3}x| > 0 \) means that the distance between \( 6 - \frac{1}{3}x \) and 0 is greater than 0. This implies that \( 6 - \frac{1}{3}x \) cannot be zero.

Step 2: Set up the Inequality

To solve the inequality \( |6 - \frac{1}{3}x| > 0 \), we need to find when \( 6 - \frac{1}{3}x \) is not equal to zero. We set up the equation: \( 6 - \frac{1}{3}x eq 0 \).

Step 3: Solve for x

Solve \( 6 - \frac{1}{3}x eq 0 \). Start by isolating \( \frac{1}{3}x \): \[ 6 eq \frac{1}{3}x \] Multiply both sides by 3 to clear the fraction: \[ 18 eq x \] Thus, \( x eq 18 \).

Step 4: Conclusion

The inequality \( |6 - \frac{1}{3}x| > 0 \) holds true whenever \( x eq 18 \). Therefore, all real numbers except 18 satisfy the inequality.

Key Concepts

Solving Inequalities

When we are approached with inequalities, our main goal is to determine the range of values that satisfy a given condition. In our given exercise, we have to solve the inequality \(|6 - \frac{1}{3}x| > 0\). This implies finding all real numbers where the condition holds.

• First, it's important to understand that an inequality involving absolute values often presents two potential scenarios. However, since the expression inside the absolute value is strictly greater than zero, it only indicates that this expression cannot be zero.
• To solve, we rewrite the condition excluding the case where the expression equals zero, i.e., 6 - \(\frac{1}{3} x \) should not be zero.
• This simplification helps us pinpoint the values that are valid, leading us to solve a simpler equation: \(6 - \frac{1}{3}x eq 0\).

By approaching inequalities this way, we can effectively determine the possible range for our variable \(x\). This analysis is key in tackling more complex problems in college-level algebra too.

Absolute Value Equations

Absolute value equations require that we view the provided expression as a distance from zero. In our problem, the purpose of the absolute value \(|6 - \frac{1}{3}x|\) is to show how far the expression is from zero.

• Understanding absolute value is crucial, as it ensures the output is always non-negative.
• By focusing on the condition \(|6 - \frac{1}{3}x| > 0\), we realize that the expression cannot be zero since absolute values depict a non-zero distance as positive.

This directs us to determine when the content within the absolute value is not zero, reinforcing that any solution must prevent this from happening. Absolute value problems often appear tricky at first glance, but deconstructing them simplifies our approach greatly.

College Algebra

In college algebra, understanding the manipulation of inequalities and absolute value equations is foundational. We often encounter more complex versions of these fundamental concepts, configured in leading to exciting problem-solving techniques.

• The inequality \(|6 - \frac{1}{3}x| > 0\) serves as a foundational problem, highlighting a core idea that is explored deeply in college algebra courses: examining and breaking down expressions.
• This problem equips us with the methodical approach to analyze and dissect seemingly complex expressions, focusing on key steps such as setting inequalities correctly and solving basic equations.
• College algebra trains us to develop critical thinking and situational analysis, identifying subtle distinctions between equations and inequalities, and their real-world implications.

Grappling with these topics not only enhances mathematical intuition but also prepares students for advanced mathematical studies and practical applications.