Problem 22 · 2019 Math Kangaroo
Stretch
Algebra & Patterns
caseworksubstitution
What is the set of all values of the parameter \(a\) for which the equation \(2 - |x| = ax\) has exactly two solutions?
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Answer: B — \(\left]-1;\,1\right[\)
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Hint 1 of 2
Graph \(y = 2 - |x|\), a tent peaking at \((0,2)\), against the line \(y = ax\) through the origin.
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Hint 2 of 2
Find which slopes \(a\) make the line meet the tent in exactly two points.
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Approach: graph the tent function against a line through the origin
- \(y = 2 - |x|\) is a tent with peak \((0,2)\) and zeros at \(x = \pm 2\); its two sides have slopes \(+1\) (left) and \(-1\) (right).
- On the right branch \(2 - x = ax\) gives \(x = \dfrac{2}{1+a}\), which is a valid solution only when \(a > -1\).
- On the left branch \(2 + x = ax\) gives \(x = \dfrac{2}{a-1}\), valid only when \(a < 1\); both branches give a solution exactly when \(-1 < a < 1\).
- Answer (B) \(\left]-1;\,1\right[\).
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