Math Problem Statement
Solution
To determine which inequality is satisfied for any real , let’s analyze each option:
a)
The expression is always positive for any real , and greater than 1. Therefore, the logarithm is always positive.
b)
This is a quadratic inequality. Solving gives roots at and . The parabola opens downwards, so it is non-positive outside the interval , not for all .
v)
This inequality does not hold for all . For , which is not greater than or equal to 2.
г)
Since , the function is always greater than 1 for any real .
д)
The expression is always positive for any real because the discriminant is negative, indicating no real roots and a parabola opening upwards.
Thus, both options a) and д) are satisfied for every real .
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Math Problem Analysis
Mathematical Concepts
Logarithms
Quadratic Equations
Absolute Value
Exponential Functions
Formulas
logarithmic properties
Quadratic equation formula: ax^2 + bx + c = 0
Absolute value definition: |x| = x for x ≥ 0, and |x| = -x for x < 0
Exponential function properties: a^x > 0 for all a > 0
Theorems
Logarithm positivity for bases greater than 1
Discriminant theorem for quadratic equations
Properties of absolute value
Growth of exponential functions
Suitable Grade Level
Grades 10-12
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