Question

5x+10/x-4>=0 решить неравентсво

Answer 1

Решение задания смотри на фотографии

Answer 2

To solve the inequality $5x + \frac{10}{x} - 4 \geq 0$, we can follow these steps:

Step 1: Find the critical points by setting the expression equal to zero (to find where it changes sign). Step 2: Create a number line and determine the sign of the expression in each interval. Step 3: Write the solution as an interval(s) where the expression is greater than or equal to zero.

Let's proceed with the solution:

Step 1: Find the critical points: To find the critical points, we set the expression $5x + \frac{10}{x} - 4$ equal to zero and solve for $x$:

$5x + \frac{10}{x} - 4 = 0$

To make it easier to work with, let's multiply the whole equation by $x$ to eliminate the fraction:

$5x^2 + 10 - 4x = 0$

Now, we have a quadratic equation. Let's solve it:

$5x^2 - 4x + 10 = 0$

To solve the quadratic equation, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 5$, $b = -4$, and $c = 10$.

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 5 \cdot 10}}{2 \cdot 5}$ $x = \frac{4 \pm \sqrt{16 - 200}}{10}$ $x = \frac{4 \pm \sqrt{-184}}{10}$

Since the discriminant $(-184)$ is negative, there are no real solutions. Therefore, there are no critical points for this inequality.

Step 2: Create a number line:

Since there are no critical points, we can choose any test points in each interval and determine the sign of the expression $5x + \frac{10}{x} - 4$ in those intervals.

Let's choose $x = 0$ and $x = 5$ as test points.

Step 3: Determine the sign of the expression in each interval:

Consider the intervals: $x < 0$, $0 < x < 5$, and $x > 5$.

For $x = -1$:

$5(-1) + \frac{10}{-1} - 4 = -5 - 10 - 4 = -19 < 0$

For $x = 1$:

$5(1) + \frac{10}{1} - 4 = 5 + 10 - 4 = 11 > 0$

For $x = 6$:

$5(6) + \frac{10}{6} - 4 = 30 + \frac{10}{6} - 4 > 0$

Step 4: Write the solution:

From the number line, we can see that the expression $5x + \frac{10}{x} - 4$ is greater than or equal to zero in the interval $x < 0$ and in the interval $x > 5$.

Thus, the solution to the inequality is:

$x < 0 \text{ or } x > 5$