Problem 62 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The inequality \(|8x-4|<0\) has no solution.

Step 1: Understand the Property of Absolute Values

Recognize that the absolute value expression \(|a|\) is always non-negative, meaning it can never be less than zero. Therefore, the inequality \(|8x-4|<0\) has no solutions since \(|8x-4|\) cannot be negative.

Step 2: Conclude that the inequality is not solvable

Since an absolute value expression cannot be less than zero, the given inequality has no solutions. For any real number substitute for \(x\), \(|8x-4|\) will always be greater than or equal to zero.

Key Concepts

Understanding the Non-Negative Property of Absolute Values

Absolute values are a fundamental concept in mathematics, often misunderstood due to their unique property. When dealing with absolute values, one must recognize the non-negative property. This means - Whenever you see an expression in absolute value symbols, such as \(|a|\), the outcome will always be a non-negative number.- Regardless of whether the expression inside the absolute value is positive, negative, or zero, the absolute value of that expression is never negative. For example, - \(|8x-4|\), regardless of the value of \(- 8x-4\), is always \(\geq 0\). This property is crucial in understanding why some absolute value inequalities, like \(|8x-4| < 0\), have no solution.

When an Inequality Has No Solution

In mathematics, an inequality may sometimes conclude with no possible solution. This often occurs in absolute value inequalities with conditions that can't be satisfied. Consider the inequality \(|8x-4| < 0\):- Since an absolute value cannot be negative by its non-negative property, \(|8x-4|\) will not meet this condition.- Thus, there aren't any values of \(x\) that will make this inequality true.Situations where the mathematical conditions given contradict the properties of the numbers involved, such as absolute values, result in no solution.Simply put, if you find an inequality suggesting an absolute value to be less than zero, you need to recognize it as unsolvable under the real number system.

Exploring Absolute Value Inequalities

Inequalities express a range of values rather than a single value. When working with absolute values in inequalities, remember: - Absolute value inequalities can have solutions if they set a condition that an absolute value is greater than, less than, or equal to a certain positive number. - But, when they ask for an absolute value to be less than zero, that's impossible. When solving absolute value inequalities, you usually aim to find values that make the expressions inside the absolute value follow the inequality condition. This involves splitting the absolute value into two possible cases, catering to both the positive and negative scenarios inside the absolute value. However, if your condition starts by requiring the absolute value to be less than zero, you don’t proceed further. Recognizing such unsolvable inequalities can save time and efforts, keeping you confident in the consistency of your mathematical abilities.