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A lion roams in the region defined by the equation |y| - x + |x| = 0. On which of the following curves can a person travel so as never to meet the lion (i.e. the curve must not intersect the lion's region)?

1. (A) y = e^(-|x|)
2. (B) y = 1/x
3. (C) y = signum(x)
4. (D) y = -|4 + |x||

Correct answer: (D) y = -|4 + |x||

Solution

For x >= 0: |x| = x so |y| - x + x = |y| = 0, meaning y = 0 (positive x-axis). For x < 0: |x| = -x so |y| - x - x = |y| - 2x = 0, i.e. |y| = 2x; but x < 0 makes the RHS negative, impossible, so no points there. Thus the lion's region is the ray y = 0, x >= 0. A safe curve must avoid this ray. y = -|4 + |x|| is always <= -4, never touching y = 0, so it is safe.

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