Bunuel wrote:
If x is an integer and \(|x - 3| + |x + 4| \leq |x + 8|\), what is the value of x ?
(1) x < 1
(2) |x| > x
M36-88
Official Solution:If \(x\) is an integer and \(|x - 3| + |x + 4| \leq |x + 8|\), what is the value of \(x\) ? The critical points (aka key points or transition points) are -8, -4 and 3 (the values of \(x\) for which the expressions in the absolute values become 0).
So, we should consider the following ranges:
If \(x < - 8\), then \(x - 3 < 0\), \(x + 4 < 0\) and \(x + 8 < 0\), so \(|x - 3| = -(x-3)\), \(|x + 4| = -(x + 4)\) and \(|x + 8| = -(x + 8)\). Thus in this range \(|x - 3| + |x + 4| \leq |x + 8|\) becomes \(-(x - 3) - (x + 4) \leq -(x + 8)\). This gives \(x \geq 7\). Since we consider \(x < - 8\) range, then for this range given inequality does not have a solution
If \(-8 \leq x \leq - 4\), then \(x - 3 < 0\), \(x + 4 \leq 0\) and \(x + 8 \geq 0\), so \(|x - 3| = -(x-3)\), \(|x + 4| = -(x + 4)\) and \(|x + 8| = x + 8\). Thus in this range \(|x - 3| + |x + 4| \leq |x + 8|\) becomes \(-(x - 3) - (x + 4) \leq x + 8\). This gives \(x \geq -3\). Since we consider \(-8 \leq x \leq - 4\) range, then for this range given inequality does not have a solution
If \(-4 < x \leq 3\), then \(x - 3 \leq 0\), \(x + 4 > 0\) and \(x + 8 > 0\), so \(|x - 3| = x-3\), \(|x + 4| = -(x + 4)\) and \(|x + 8| = x + 8\). Thus in this range \(|x - 3| + |x + 4| \leq |x + 8|\) becomes \(-(x - 3) + (x + 4) \leq x + 8\). This gives \(x \geq -1\). Since we consider \(-4 < x \leq 3\) range, then for this range given inequality gives the following integer solutions: -1, 0, 1, 2, and 3.
If \(x > 3\), then \(x - 3 > 0\), \(x + 4 > 0\) and \(x + 8 > 0\), so \(|x - 3| = x-3\), \(|x + 4| = x + 4\) and \(|x + 8| = x + 8\). Thus in this range \(|x - 3| + |x + 4| \leq |x + 8|\) becomes \(x - 3 + x + 4 \leq x + 8\). This gives \(x \leq 7\). Since we consider \(x > 3\) range, then for this range given inequality gives the following integer solutions: 4, 5, 6, and 7.
So, \(|x - 3| + |x + 4| \leq |x + 8|\) holds true for the following integer values of \(x\): -1, 0, 1, 2, 3, 4, 5, 6, and 7.
(1) \(x < 1\)
\(x\) can be 0 or -1. Not sufficient.
(2) \(|x| > x\)
The above implies that \(x\) is negative, therefore \(x=-1\). Sufficient.
Answer: B
Can we see if the ranges are valid by picking a number from the range and substituting it in the inequality?