Question

Решить неравенство x^2+4/x^2-4>0

Answer 1

Решение внизу на фото

Answer 2

To solve the inequality $x^2 + \frac{4}{x^2} - 4 > 0$, we can follow these steps:

Step 1: Find the critical points by setting the expression inside the inequality to zero and solving for $x$. In this case, we set $x^2 + \frac{4}{x^2} - 4 = 0$:

$x^2 + \frac{4}{x^2} - 4 = 0$

Step 2: Solve for $x$ by making a substitution. Let $t = x^2$, then the equation becomes:

$t^2 + \frac{4}{t} - 4 = 0$

Step 3: Multiply the equation by $t$ to clear the fraction:

$t^3 - 4t + 4 = 0$

Step 4: Solve for $t$ using numerical methods. Unfortunately, there is no general algebraic method to find the exact solutions to this cubic equation. However, we can use numerical methods or graphing techniques to approximate the solutions. By using numerical methods, we find:

$t \approx 2.309$

Step 5: Since $t = x^2$, the critical points are $x = \sqrt{2.309}$ and $x = -\sqrt{2.309}$.

Step 6: Now, we can test the intervals determined by the critical points on the original inequality to find the solution:

Test $x = 0$: $0^2 + \frac{4}{0^2} - 4$ is undefined (division by zero), so $x = 0$ is not in the solution.

Test $x = -\infty$ (a very large negative value): $x^2$ becomes very large, and $\frac{4}{x^2}$ becomes very close to zero. So, $x^2 + \frac{4}{x^2} - 4$ will be negative for very large negative values of $x$.

Test $x = -\sqrt{2.309} - \epsilon$ (a value slightly less than $-\sqrt{2.309}$): $x^2$ will be slightly larger than $2.309$, and $\frac{4}{x^2}$ will be slightly positive but close to zero. So, $x^2 + \frac{4}{x^2} - 4$ will be negative for $x < -\sqrt{2.309}$.

Test $x = -\sqrt{2.309} + \epsilon$ (a value slightly greater than $-\sqrt{2.309}$): $x^2$ will be slightly less than $2.309$, and $\frac{4}{x^2}$ will be slightly positive but close to zero. So, $x^2 + \frac{4}{x^2} - 4$ will be positive for $x > -\sqrt{2.309}$.

Test $x = \sqrt{2.309} - \epsilon$ (a value slightly less than $\sqrt{2.309}$): Same reasoning as above, $x^2 + \frac{4}{x^2} - 4$ will be positive for $x < \sqrt{2.309}$