Non-Positive and Non-Negative Numbers

_N_ is a two-digit positive integer whose digits are different, and _M_ is a two-digit negative integer whose digits are the same. Which of the following is always false?

Incorrect. [[snippet]] This is true when _N_ and _N_ + _M_ have different signs. This will happen whenever _M_ is farther from zero than _N_, because in this case _N_ + _M_ is negative. Let _N_ = 10 and _M_ = -11. Then _N_ (_N_ + _M_) = 10(-1) = -10, which is non-positive.

Correct! _M_ - _N_ attains its greatest possible value when _M_ = -11 and _N_ = 10. In this case _M_ - _N_ = -21, since -11 - 10 = -21. This implies the greatest possible value for _M_ - _N_ + 20 is -1. The fact that _M_ - _N_ + 20 cannot be greater than -1 guarantees that _M_ - _N_ + 20 cannot be non-negative, and so the statement is always false.

Incorrect. [[snippet]] If _M_ = -11 and _N_= 10, then _M_ - _N_ = -21, which is non-positive. _M_ - _N_ is actually non-positive for any choice of _M_ and _N_.

Incorrect. [[snippet]] _N_ + _M_ can certainly be non-negative. This will happen whenever _N_ is farther from zero than _M_. For example, if _N_ = 98 and _M_ = -88, then _N_ + _M_ = 98 - 88 = 10.

Incorrect. [[snippet]] If _N_ = 98 and _M_ = -11, then _N_ + _M_ - 87 = 0, which is non-negative. Since the statement is true for this case, it is not always false.

_N_ (_N_ + _M_) is non-positive

_N_ + _M_ - 87 is non-negative

_M_ - _N_ + 20 is non-negative

_M_ - _N_ is non-positive

_N_ + _M_ is non-negative