Math Problem Statement

Кое от следните неравенства е изпълнено за всяко реално x?

Solution

To determine which inequality is satisfied for any real xx, let’s analyze each option:

a) log2(x2+1)>0\log_2 (x^2 + 1) > 0

The expression x2+1x^2 + 1 is always positive for any real xx, and greater than 1. Therefore, the logarithm is always positive.

b) x2+x+20-x^2 + x + 2 \leq 0

This is a quadratic inequality. Solving x2+x+2=0-x^2 + x + 2 = 0 gives roots at x=1x = -1 and x=2x = 2. The parabola opens downwards, so it is non-positive outside the interval [1,2][-1, 2], not for all xx.

v) x1x|x - 1| \geq x

This inequality does not hold for all xx. For x=2x = 2, 21=1|2 - 1| = 1 which is not greater than or equal to 2.

г) 33x+1>13^{3x+1} > 1

Since 33x+1=333x3^{3x+1} = 3 \cdot 3^{3x}, the function is always greater than 1 for any real xx.

д) x2+x+2>0x^2 + x + 2 > 0

The expression x2+x+2x^2 + x + 2 is always positive for any real xx because the discriminant b24ac=18=7b^2 - 4ac = 1 - 8 = -7 is negative, indicating no real roots and a parabola opening upwards.

Thus, both options a) and д) are satisfied for every real xx.

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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