Question

4*(4^-x)-9*(2^-x)+2>0

Answer 1

Решение приложено к снимку:

Answer 2

4*(4^-x)-9*(2^-x)+2>0
4*(2^-x) ²  -9*(2^-x) +2>0
замена (2^-x)= а      ОДЗ а>0
4a²  -9a +2>0
D=81-32=49
a₁ =(9+7)/8=2                  2⁻ˣ =2     x₁= -1
a₂ =(9-7)/8= 1/4= 2⁻²       2⁻ˣ = 2⁻²  x₂= 2
      +                        -             +
____________-1 ____ 2_____________

x∈(-∞;-1)∪(2;+∞)

Answer 3

We can simplify this inequality by first noticing that both terms involve exponential functions with negative exponents. Let's rewrite the equation using a common base:

4*(2^(-2x))-9*(2^(-x))+2 > 0

Next, we can substitute a variable for 2^(-x), say y = 2^(-x). Then, we have:

4y^2 - 9y + 2 > 0

This is now a quadratic inequality. We can solve for the roots of the quadratic equation:

4y^2 - 9y + 2 = 0

Using the quadratic formula, we get:

y = (9 ± sqrt(65)) / 8

Since y = 2^(-x), we can solve for x by taking the logarithm of both sides:

x = -log2(y)

Therefore, the solutions for the inequality are:

x < -log2[(9 + sqrt(65)) / 8] ≈ -0.543

or

x > -log2[(9 - sqrt(65)) / 8] ≈ 2.121

So the solution set is:

{x | x < -0.543 or x > 2.121}