Question

Решить неравенство 1/х^2>1

Answer 1

Решение прикрепляю ниже

Answer 2

To solve the inequality $\frac{1}{x^2} > 1$, follow these steps:

Step 1: Identify the domain. Since the inequality involves a fraction with $x$ in the denominator, we need to ensure that $x$ is not equal to 0 to avoid division by zero. Thus, the domain is $x \neq 0$.

Step 2: Get rid of the fraction. To eliminate the fraction, multiply both sides of the inequality by $x^2$ (which is positive since $x \neq 0$). Be cautious here because multiplying by a negative number would reverse the inequality.

$\frac{1}{x^2} \cdot x^2 > 1 \cdot x^2$

Step 3: Simplify the inequality. On the left side, the $x^2$ terms cancel out, leaving us with:

$1 > x^2$

Step 4: Move all terms to one side. To further solve the inequality, let's move all terms to one side of the inequality:

$x^2 - 1 < 0$

Step 5: Factorize and find critical points. Now, factorize the left side:

$(x + 1)(x - 1) < 0$

The critical points are where the inequality sign might change. In this case, the critical points are $x = -1$ and $x = 1$.

Step 6: Test intervals. To determine the solution intervals, create a sign chart or use test points within each interval.

For $x < -1$, choose $x = -2$ (test point): $((-2 + 1)(-2 - 1) < 0)$ which simplifies to $(3) < 0$, which is true.

For $-1 < x < 1$, choose $x = 0$ (test point): $(0 + 1)(0 - 1) < 0$ which simplifies to $(-1) < 0$, which is true.

For $x > 1$, choose $x = 2$ (test point): $(2 + 1)(2 - 1) < 0$ which simplifies to $(3) < 0$, which is false.

Step 7: Finalize the solution. The solution to the inequality is:

$x < -1$ or $0 < x < 1$

Remember that $x \neq 0$ because it is excluded from the domain due to the initial fraction $\frac{1}{x^2}$.