Problem 140 - College Algebra Essentials

Question

Explain why \(|x|<-4\) has no solution.

Step-by-Step Solution

Answer

The expression \(|x|<-4\) has no solution because the absolute value of a number is non-negative, meaning it can never be less than a negative number.

Step 1: Understanding the Absolute Value

Firstly, to analyse the equation \(|x|<-4\), it's important to understand what an absolute value signifies. Absolute value, denoted by \(|x|\), of a number refers to the distance of that number from zero on the number line, regardless of the direction. Therefore, it is always non-negative.

Step 2: Analyzing the Combining of Absolute Value and Negation

Next, consider the requirement that the absolute value of \(x\) is less than -4. Because the absolute value of a number is always non-negative, it cannot be less than any negative number, including -4.

Step 3: Summarizing the Conclusion

From the above exploration, it is clarified that an absolute value can't be less than a negative number. Therefore, the mathematical expression \(|x|<-4\) has no solution in the realm of real numbers.