Non-Positive and Non-Negative Numbers

The variables _w_, _x_, _y_, and _z_ are integers with the following properties: (i) _w_ and _x_ are non-negative. (ii) _y_ and _z_ are non-positive. (iii) _w_ = _z_. (iv) _w_ = _x_ - 1 and _y_ = _z_ - 1. Which of the following must be true?

Incorrect. [[snippet]] The value of _w_ cannot be equal to -4 because _w_ is defined as non-negative.

Correct! To demonstrate that this must be true, you can find the actual values of _w_, _x_, _y_, and _z_. Notice that _w_ = _z_, but _w_ is non-negative, and _z_ is non-positive. The only way they can be equal to one another is for _w_ = _z_ = 0, since 0 is the only number that is both non-positive and non-negative. As _w_ = 0, you know that _x_ = 1 by property (iv). Also, by property (iv), you know that _y_ = -1. You have now determined that _w_ = 0, _x_ = 1, _y_ = -1, and _z_ = 0. It is now clear that the only answer choice that is correct is _w_ + _x_ + _y_ + _z_ = _wz_, which you can verify by __Plugging In__ the values.

Incorrect. [[snippet]] If _x_ = 2, then property (iv) implies that _w_ = 1. Yet, this means that _w_ cannot be equal to _z_, since _z_ is non-positive. Therefore, this answer choice violates property (iii).

Incorrect. If _w_ - _z_ = 1, then _w_ = _z_ + 1. Yet, the assumptions state that _w_ = _z_. This answer choice violates property (iii).

Default step content.

Incorrect. [[snippet]] If _y_ = -3, then property (iv) specifies that _z_ = -2. Yet, this means that _z_ cannot be equal to _w_, since _w_ is non-negative. Therefore, this answer choice violates property (iii).

_w_ = -4

_y_ = -3

_x_ = 2

_w_ + _x_ + _y_ + _z_ = _wx_

_w_ - _z_ = 1