Problem 27 · 2018 Math Kangaroo
Stretch
Algebra & Patterns
How many real solutions does the equation \(\bigl\lvert\,\lvert 4^x - 3\rvert - 2\,\bigr\rvert = 1\) have?
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Answer: B — 3
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Hint 1 of 2
Peel the absolute values from the outside in.
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Hint 2 of 2
Each step splits into two cases; keep only those with a positive power of 4.
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Approach: unpeel nested absolute values and count valid exponentials
- ||4^x−3|−2| = 1 means |4^x−3|−2 = ±1, so |4^x−3| = 3 or 1.
- |4^x−3| = 3 gives 4^x = 6 (valid) or 4^x = 0 (impossible).
- |4^x−3| = 1 gives 4^x = 4 or 4^x = 2, both valid.
- Three positive values of 4^x (6, 4, 2) give 3 real solutions.
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