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Assignment:Find the solution set of inequality e{(lnx/sqrtx)+(lnx/(x^(1/8)))}-1 >1-e{lnx/sqrtx-lnx/(x^(1/8))} .
Question type: Inequality
Solution:
The inequality can be reduced to 1 inequality:
e ( ( ln x / sqrt x ) + ( ln x / ( x ^ ( 1 / 8 ) ) ) ) - 1 >1 - e ( ln x / sqrt x - ln x / ( x ^ ( 1 / 8 ) ) ) (1)
From the definition field of ln
x > 0 (2 )
From the definition field of √
x ≥ 0 (3 )
From the definition field of ln
x > 0 (4 )
From the definition field of divisor
x ^ ( 1 / 8 ) ≠ 0 (5 )
From the definition field of ln
x > 0 (6 )
From the definition field of √
x ≥ 0 (7 )
From the definition field of ln
x > 0 (8 )
From the definition field of divisor
x ^ ( 1 / 8 ) ≠ 0 (9 )
From inequality(1):
x > 1
From inequality(2):
x > 0
From inequality(3):
x ≥ 0
From inequality(4):
x > 0
From inequality(5):
x ∈ R (R为全体实数),即在实数范围内,不等式恒成立!
From inequality(6):
x > 0
From inequality(7):
x ≥ 0
From inequality(8):
x > 0
From inequality(9):
x ∈ R (R为全体实数),即在实数范围内,不等式恒成立!
From inequalities (1) and (2)
x > 1 (10)
From inequalities (3) and (10)
x > 1 (11)
From inequalities (4) and (11)
x > 1 (12)
From inequalities (5) and (12)
x > 1 (13)
From inequalities (6) and (13)
x > 1 (14)
From inequalities (7) and (14)
x > 1 (15)
From inequalities (8) and (15)
x > 1 (16)
From inequalities (9) and (16)
x > 1 (17)
The final solution set is :
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