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1). there are five elements so the median has to be the middle number, therefore the middle number would have to be zero, so x would have to be zero, however zero is not positive or negative.

Don`t think im understanding the question quite right??

1). there are five elements so the median has to be the middle number, therefore the middle number would have to be zero, so x would have to be zero, however zero is not positive or negative.

Don`t think im understanding the question quite right??

Thanks

The question asks to find the value of x not to determine whether it's positive or negative.

What is the value of \(x\)?

Note that the median of a set with odd number of elements is just the middle element, when arranged in ascending/descending order.

(1) The median of set \(\{x, -1, 1, 3, -x\}\) is 0. The median of this set of 5 (odd) elements must be the middle term, hence \(x=0\). Sufficient.

(2) The median of set \(\{x, -1, 1, 3, -x\}\) is \(\frac{x}{2}\). It could be that \(x=0\) (in this case the set is {-1, 0, 0, 1, 3}) as well as it could be that \(x=2\) (in this case the set is {-2, -1, 1, 2, 3}). Not sufficient.

But they label that x is negative and positive, and zero is not negative or positive, thats just what confuses me, why have they done that, or why is it irrelevant

But they label that x is negative and positive, and zero is not negative or positive, thats just what confuses me, why have they done that, or why is it irrelevant

You can write 0 as -0, there is nothing wrong with it (if it's what you are talking about).
_________________

Median of a set is the middle value when the numbers are arranged in increasing order. When there are odd number of numbers e.g. 5 here, the median is simply the 3rd number.

Arranging the set in increasing order: (-1, 1, 3) Now there are two numbers x and -x too in this set which we don't know where to place right now.

1. The median of set (x, 1, -1, 3, -x) is 0 We are given that the median here is 0 which means the third number must be 0. Since we don't have any 0 yet in the set, x (and hence -x too) must be 0. The set must be (-1, 0, 0, 1, 3). There is no other way to get a median of 0. Hence x = 0. Sufficient.

2. The median of set (x, 1, -1, 3, -x) is x/2 We saw above that x could be 0 in which case median will be x/2 = 0/2 = 0. So the value x = 0 satisfies this condition. The median could also be the middle number right now i.e. 1. If x = 2, we get the set (-2, -1, 1, 2, 3) and the median here is 1 which is x/2. Hence x = 2 also satisfies this condition. This means we have two values for x (0 and 2) and not a unique value so this statement alone is not sufficient.

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