Multiple choice

How many roots does the following equation possess $3^{|x|} \left { \left | 2-|x| \right | \right }=1?$

  1. $1$
  2. $2$
  3. $3$
  4. $4$
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D Correct answer
Explanation

The equation 3^|x| * |2 - |x|| = 1 can be solved by testing cases for |x|. Graphing or testing values shows intersections at |x| = 1, |x| = 2, and two other points, totaling 4 roots.

AI explanation

The equation is 3^|x| * ||2 - |x|| = 1, which implies ||2 - |x|| = 3^(-|x|). Since 3^(-|x|) is always strictly greater than 0 and less than 1, we must have ||2 - |x|| > 0. This yields two cases for the absolute value: 2 - |x| = 3^(-|x|) or |x| - 2 = 3^(-|x|). By analyzing the intersections of these continuous functions, the first case yields two symmetric roots and the second case also yields two symmetric roots. Thus, there are exactly four real roots for the equation.

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