Special Case Solutions

Property If an isolated absolute value expression equals 0, there is only one solution. If it equals a negative number, there is no solution, written as $\emptyset$. Explanation Watch out for these special scenarios! If your isolated absolute value equals zero, you've hit the jackpot with just one answer. But if it equals a negative number, stop right there! It's a mathematical trap. Since absolute value represents distance, it can't be negative, so there's no solution at all. An empty treasure chest! Examples Solve $\frac{7|x|}{3} + 5 = 5$. Isolating gives $|x|=0$. The only solution is $x=0$. Solve $2|x| + 12 = 4$. Isolating gives $2|x|= 8$, so $|x|= 4$. This is impossible, so the solution is no solution, $\emptyset$.