Problem 140 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expression \(|x|<-4\) possesses no solutions because the absolute value of a number is always non-negative—it cannot be less than zero, let alone a negative number such as -4.

Step 1: Understand absolute value

An absolute value of a number \(x\), denoted as \(|x|\) is the distance of \(x\) from zero on the number line, and it's always non-negative, i.e., \(|x| \geq 0\).

Step 2: Analyze the inequality

The inequality \(|x|<-4\) implies that the absolute value of \(x\) is less than -4. However, from the definition of absolute value in Step 1, we know that absolute value is always non-negative. Therefore, there is no real number \(x\) that can satisfy this inequality.

Step 3: Conclusion

Since there is no real number \(x\) that can make the absolute value less than -4, the inequality \(|x|<-4\) has no solution.