Floor Ceiling Ceiling Floor

Algebra Level 3

If the range of positive xx satisfy the equation xx+xx=111 \lceil x \lfloor x \rfloor \rceil + \lfloor x \lceil x \rceil \rfloor = 111 is αxβ \alpha \leq x \leq \beta . What is the value of 8α+7β8\alpha + 7 \beta?

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