Solve \( |x|+7<4 \) \( \{x \mid x<-11 \) or \( x>-3\} \) \( \{x \mid-3

Ask by Rodriguez Olson. in the United States

Feb 25,2025

To solve the inequality \( |x| + 7 < 4 \), we first isolate the absolute value: \[ |x| < 4 - 7 \] This simplifies to: \[ |x| < -3 \] Since the absolute value cannot be less than a negative number, there are no values of \( x \) that satisfy this inequality. Thus, the solution set is \( \varnothing \) (the empty set). If you ever find yourself tangled up in inequalities involving absolute values, remember: the absolute value of a number speaks to its distance from zero. When that distance is said to be less than a negative number, you're essentially saying, "I want something to be smaller than nothing," which is a physical impossibility! Additionally, it can be helpful to visualize absolute value inequalities on a number line, separating the positive and negative cases. Here, knowing that the absolute value is intrinsically non-negative leads you to a swift conclusion of an empty solution set—simplicity at its finest!