Question

Решите неравенство 6x^4-x^2-5>0

Answer 1

Решение задания смотри на фотографии

Answer 2

To solve the inequality $6x^4 - x^2 - 5 > 0$, we can use a method called "interval testing." Here are the steps:

Step 1: Factor the left side of the inequality if possible. Step 2: Find the critical points by setting the expression inside the inequality equal to zero and solving for $x$. Step 3: Create test intervals using the critical points and any other relevant points. Step 4: Test the inequality within each interval to determine the solution.

Let's go through each step:

Step 1: The inequality $6x^4 - x^2 - 5 > 0$ is already in a simplified form, so there's no need to factor it.

Step 2: Find the critical points by setting the expression inside the inequality equal to zero and solving for $x$:

$6x^4 - x^2 - 5 = 0$

This is a quadratic equation in terms of $x^2$. Let's solve for $x^2$:

$x^2 = \frac{1 \pm \sqrt{1 + 4 \cdot 6 \cdot 5}}{2 \cdot 6} = \frac{1 \pm \sqrt{121}}{12} = \frac{1 \pm 11}{12}$

So, the two critical points are:

$x^2 = \frac{1 + 11}{12} = \frac{12}{12} = 1$ $x^2 = \frac{1 - 11}{12} = \frac{-10}{12} = -\frac{5}{6}$

Step 3: Create test intervals using the critical points and any other relevant points. We'll use the following intervals: $(- \infty, -\frac{5}{6}), (-\frac{5}{6}, 1), (1, \infty)$.

Step 4: Test the inequality within each interval to determine the solution.

In the interval $(- \infty, -\frac{5}{6})$:

Let's pick a test point, $x = -1$, and check if $6(-1)^4 - (-1)^2 - 5 > 0$:

$6(-1)^4 - (-1)^2 - 5 = 6 - 1 - 5 = 0$

Since the inequality is not satisfied in this interval, we exclude it from the solution.

In the interval $(- \frac{5}{6}, 1)$:

Let's pick a test point, $x = 0$, and check if $6(0)^4 - (0)^2 - 5 > 0$:

$6(0)^4 - (0)^2 - 5 = -5$

Since the inequality is satisfied in this interval, we include it in the solution.

In the interval $(1, \infty)$:

Let's pick a test point, $x = 2$, and check if $6(2)^4 - (2)^2 - 5 > 0$:

$6(2)^4 - (2)^2 - 5 = 96 - 4 - 5 = 87$

Since the inequality is satisfied in this interval, we include it in the solution.

Therefore, the solution to the inequality $6x^4 - x^2 - 5 > 0$ is $x \in (- \frac{5}{6}, 1) \cup (1, \infty)$.