Question 1209357: Let x, y, and z be nonzero real numbers. Find all possible values of
(x + y + z)/(|x| + |y| + |z|).
Answer by yurtman(42)   (Show Source):
You can put this solution on YOUR website! **1. Consider the Sign Combinations**
* **All positive:** If x, y, and z are all positive, then:
(x + y + z) / (|x| + |y| + |z|) = (x + y + z) / (x + y + z) = 1
* **All negative:** If x, y, and z are all negative, then:
(x + y + z) / (|x| + |y| + |z|) = -(x + y + z) / (-(x + y + z)) = 1
* **Two positive, one negative:**
* Let's say x and y are positive, and z is negative:
(x + y + z) / (|x| + |y| + |z|) = (x + y - |z|) / (x + y + |z|)
This value will be between 0 and 1, depending on the relative magnitudes of x, y, and z.
* **Two negative, one positive:**
* Similar to the previous case, the value will be between -1 and 0.
* **One positive, two negative:**
* Similar to the previous cases, the value will be between -1 and 0.
**2. Determine the Possible Values**
* Based on the sign combinations, the possible values of (x + y + z) / (|x| + |y| + |z|) range from **-1 to 1**, inclusive.
**Therefore, the possible values of (x + y + z) / (|x| + |y| + |z|) are all real numbers in the interval [-1, 1].**
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