True or false :
For all real x , the inequality x < x holds true.
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The inequality x < x only holds true for real values x > 1
Eg. x = 4 ⟹ x = 4 = 2 < 4
If x = 1 or x = 0 , we have x = x :
x = 1 ⟹ x = 1 = 1 = x
x = 0 ⟹ x = 0 = 0 = x
And for real values 0 < x < 1 , the inequality becomes x > x
Eg. x = 0 . 0 1 ⟹ x = 0 . 0 1 = 0 . 1 > 0 . 0 1
Lastly, for real values x < 0 , we will get a complex value for its square root
Eg. x = − 4 ⟹ x = − 4 = − 1 4 = 2 i
Now, I do not know whether we can compare complex values with real values or not. Hopefully, someone else will answer this question.
Even without considering real values x < 0 , we already have sufficient evidence to prove that x < x does not hold true for all real x .
Thus, the answer is False
The inequality is false for all real x ≤ 1
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False because, for example: