Given propositions : "The equation has exactly two distinct negative roots" and : "The inequality has real number solutions". If is a true proposition and is a false proposition, find the range of real number values for .
Solution
For proposition to be true, the equation must have two distinct negative roots, which implies:
This simplifies to:
> 0 \
-m 2$.
For proposition to be true, the inequality must have real number solutions, which implies . This is only possible when , so solving this inequality gives us .
Since is true and is false, and must have opposite truth values. We consider two cases:
1. is true and is false. This leads to the system of inequalities:
which has no valid values for .
2. is false and is true. This leads to the system of inequalities:
which simplifies to .
Thus, the range of real number values for is .
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