Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation or inequality. $$|8 x-4|<0$$

Step-by-Step Solution

Answer

No solution: absolute value cannot be negative.

Step 1: Analyze the absolute value inequality

The inequality is \(|8x - 4| < 0\).

By definition, the absolute value of any real number is always non-negative: \(|a| \geq 0\) for all \(a \in \mathbb{R}\).

Step 2: Determine the solution

Since \(|8x - 4| \geq 0\) for all real \(x\), there is no value of \(x\) for which \(|8x - 4| < 0\).

The solution set is \(\boxed{\emptyset}\) (no solution).

Key Concepts

Absolute Value

Absolute value is all about distance, specifically how far a number is from zero on a number line. This measurement always results in a non-negative number, no matter whether the original number is positive or negative. For instance, the absolute value of both 5 and -5 is 5. This is because each number is exactly 5 units away from zero on the number line. In mathematical terms, the absolute value of any number, say \( x \), is denoted as \( |x| \) and is defined as follows:

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \), making it a positive number.

Therefore, the absolute value helps in removing any negative sign, giving us a clear sense of distance.

Inequalities

Inequalities are mathematical statements used to compare expressions. They say one quantity is larger or smaller than another. Inequality symbols include \(<\), \(>\), \(\leq\), and \(\geq\).

• \(a < b\) means \(a\) is less than \(b\).
• \(a > b\) means \(a\) is greater than \(b\).
• \(a \leq b\) means \(a\) is less than or equal to \(b\).
• \(a \geq b\) means \(a\) is greater than or equal to \(b\).

When absolute values are involved, it’s essential to remember that they cannot be less than zero. An inequality such as \(|8x - 4| < 0\) has no solution because the absolute value expression can never be negative. This understanding helps us instantly recognize when an inequality doesn't hold.

Non-negative Numbers

Non-negative numbers are numbers that are either positive or zero. In other words, they are numbers that are not negative. This set includes all positive integers, decimals, fractions, and zero. When dealing with absolute values, we need to remember that they fall into the category of non-negative numbers. This means expressions like \(|x|\) will always result in a number that is greater than or equal to zero. Recognizing a number as non-negative is crucial when solving problems involving absolute values because it allows us to dismiss impossible inequalities quickly.

Number Line Representation

A number line is a visual representation of numbers along a straight path, marked with intervals representing integers. Each point on the number line corresponds to a number, and the further right a point is, the greater the number it represents.Absolute values can be easily understood through number lines as they represent distance from zero. The meaning can be visualized by showing the same distance towards both positive and negative directions from zero. For example, the point for \( |5|\) and \( |-5|\) would both be placed 5 units away from zero but in opposite directions.Number lines help us visually grasp concepts such as the non-negativity of absolute values and the comparison of numbers through inequalities. They serve as a helpful tool for understanding how various inequalities and equality operate within the context of absolute values.