Problem 73 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

All real numbers except 18.

Step 1: Understand the Absolute Value Inequality

The inequality given is \(|6 - \frac{1}{3}x| > 0\). This means that the expression inside the absolute value, \(6 - \frac{1}{3}x\), must be greater than 0 or less than 0. It cannot be equal to zero because it needs to be strictly greater than 0, since it is an inequality sign without an "equal to" component.

Step 2: Set Up Two Separate Inequalities

Because the absolute value is greater than zero, the expression inside the absolute value can be either positive or negative but not zero. Therefore, we will set up two inequalities: 1. \(6 - \frac{1}{3}x > 0\)2. \(6 - \frac{1}{3}x < 0\)

Step 3: Solve the First Inequality

Let's solve the first inequality: \(6 - \frac{1}{3}x > 0\).Subtract 6 from both sides:\(-\frac{1}{3}x > -6\).Multiply both sides by -3 (remember to reverse the inequality sign when multiplying or dividing by a negative number):\(x < 18\).

Step 4: Solve the Second Inequality

Now, solve the second inequality: \(6 - \frac{1}{3}x < 0\).Subtract 6 from both sides:\(-\frac{1}{3}x < -6\).Multiply both sides by -3 (again, reverse the inequality):\(x > 18\).

Step 5: Combine the Solutions

Combine the solutions from the two inequalities. From Step 3, we have \(x < 18\). From Step 4, \(x > 18\). This means all values except \(x = 18\) satisfy the inequality. Thus, the solution is all real numbers except \(x = 18\).

Key Concepts

Absolute Value Inequality

Understanding absolute value inequalities is an important skill in precalculus. Absolute value refers to the distance of a number from zero on the number line, meaning it is always non-negative. When you see an absolute value inequality like \(|6 - \frac{1}{3}x| > 0\), it means the expression inside the absolute value should not be equal to zero.
Absolute value inequalities can have different forms, such as \(>\), \(<\), \(\geq\), and \(\leq\). Each form tells us something different about the relationship between numbers. For example:

• A "greater than zero" inequality means the expression must be positive.
• A "less than zero" would not be valid since absolute values are non-negative.

For the given inequality, \(|6 - \frac{1}{3}x| > 0\), we start by acknowledging that the expressions inside the absolute value, either positive or negative, should not equal zero.

Inequality Solutions

To solve absolute value inequalities, you create two separate inequalities to cover both possible scenarios of the expression inside the absolute value. For our example, \(|6 - \frac{1}{3}x| > 0\), we set up two conditions:

• \(6 - \frac{1}{3}x > 0\)
• \(6 - \frac{1}{3}x < 0\)

This is because the absolute value inequality indicates that the expression must either be greater than zero (positive) or less than zero (negative), but definitely not zero itself. Solving these inequalities will help us find the range of values for \(x\) that satisfy the original equation.
The solution is reached when you determine the conditions under which each inequality holds, and then combine them for the overall solution. Understanding this approach is crucial for solving more complex inequalities in the future.

Algebraic Manipulations

Algebraic manipulations are the procedures carried out to isolate the variable and solve the inequality. Let's break down how to solve each inequality step by step.
For \(6 - \frac{1}{3}x > 0\):

• First, subtract 6 from both sides to simplify: \(-\frac{1}{3}x > -6\).
• Next, multiply both sides by -3 to solve for \(x\). Remember, multiplying by a negative reverses the inequality sign, giving \(x < 18\).

For the second inequality, \(6 - \frac{1}{3}x < 0\):

• Subtract 6 from both sides: \(-\frac{1}{3}x < -6\).
• Again, multiply both sides by -3, reversing the inequality, so \(x > 18\).

Combining both inequalities, we deduce that \(x\) can be any number except 18, emphasizing the comprehensive nature of algebraic manipulations to find all valid solutions.