Question

The equation | | x + 3 | - 2 | = p, where p is...

The equation | | x + 3 | - 2 | = p, where p is a constant integer has exactly three distinct solutions. Find the value of p.

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Answer #1

If | | x + 3 | - 2 | = p, then p = ± | x + 3 | - 2. Now, there are 3 possibilities:

Case I:

x > -3 and p =| x + 3 | - 2.   Then x+3 ≥ 0 so that p = (x+3)-2 = x +1.

Case II:

x > -3 and p = -| x + 3 | - 2.   Then x+3 ≥ 0 so that p = - (x+3)-2 = -x-5.

Case III:

x < -3 and p = -| x + 3 | - 2 . Then p = (x+3)-2 = x +1 which is same as in case I.

Case IV:

x < -3 and p = | x + 3 | - 2 . Then p = -(x+3) -2 = -x-5 which is same as in Case II.

Case V:

x = -3. Then | x + 3 | = 0 so that p = -2.

Thus, there are three distinct solutions as under:

1. If x > -3 , and p =| x + 3 | - 2 OR, if p = -| x + 3 | - 2. Then p = x +1.

2. If x > -3, and p = -| x + 3 | - 2 OR, if x < -3 and p = | x + 3 | - 2. Then p =-x-5

3. If x = -3. Then p = -2.

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