Final answer:
To answer how many solutions the equation ³|x|=|2-|x|| has, we must examine different cases based on the properties of absolute values. Unfortunately, the equation is possibly misrepresented due to typos. Proper interpretation is required for solving.
Step-by-step explanation:
To solve the mathematical problem completely, we need to find the number of solutions to the equation ³|x|=|2-|x||. Here we recognize that solving absolute value equations often involves considering multiple cases, since the absolute value function can return the same value for a positive and a negative number. Unfortunately, there seems to be some confusion in the typographical representation of the equation provided; the operation ³ could either denote a cube root or an imperfect representation of a different mathematical procedure. Assuming this is a typo and interpreting the equation attempt correctly is essential for arriving at a solution.
We can start by noting that, typically, |x| will have two solutions: x and -x. The equation &em;|2-|x||&em; therefore is also an absolute value equation and will need to be considered in two cases: when 2-|x| ≥ 0 and when 2-|x| < 0. This gives us two different scenarios to consider. Each case will then be solved individually and the number of solutions from these cases will give us the total number of solutions for the original equation.
It's important to recognize that sometimes, the solutions obtained might not be valid for the original equation, or there may be duplicated solutions amongst the cases.