Question

(2x-2,5)^2 (3х-13)^3 <0;

Answer 1

Ответ:

Объяснение:

(2x-2,5)^2 (3х-13)^3 <0;

(2x-2,5)^2>0 всегда полажител; значить

(3х-13)^3 <0; и (3х-13) <0;   3х<13; x<13/3;

x∈(-∞; 13/3)

Answer 2

To solve the inequality $(2x - 2.5)^2 (3x - 13)^3 < 0$, you can follow these steps:

1. Find the critical points where the expression equals zero by setting each factor inside the parentheses equal to zero and solving for $x$:

   $(2x - 2.5)^2 = 0$ and $(3x - 13)^3 = 0$.

   Solve each equation separately:

   For $(2x - 2.5)^2 = 0$:

   $2x - 2.5 = 0$ $2x = 2.5$ $x = 1.25$

   For $(3x - 13)^3 = 0$:

   $3x - 13 = 0$ $3x = 13$ $x = \frac{13}{3}$

   So, the critical points are $x = 1.25$ and $x = \frac{13}{3}$.

2. Now, you need to determine the sign of each factor in the inequality in the intervals created by the critical points.

   Let's examine three intervals: $x < \frac{13}{3}$, $\frac{13}{3} < x < 1.25$, and $x > 1.25$.

   a) For $x < \frac{13}{3}$:

   • $(2x - 2.5)^2$ is positive because it's the square of a real number.
   • $(3x - 13)^3$ is also positive because it's the cube of a real number.

   Therefore, the product $(2x - 2.5)^2 (3x - 13)^3$ is positive in this interval.

   b) For $\frac{13}{3} < x < 1.25$:

   • $(2x - 2.5)^2$ is positive.
   • $(3x - 13)^3$ is negative because $(3x - 13)$ is negative in this interval, and cubing a negative number results in a negative value.

   Therefore, the product $(2x - 2.5)^2 (3x - 13)^3$ is negative in this interval.

   c) For $x > 1.25$:

   • $(2x - 2.5)^2$ is positive.
   • $(3x - 13)^3$ is also positive.

   Therefore, the product $(2x - 2.5)^2 (3x - 13)^3$ is positive in this interval.

3. To summarize the results:

   • The inequality $(2x - 2.5)^2 (3x - 13)^3 < 0$ is satisfied in the interval $\frac{13}{3} < x < 1.25$.

So, the solution to the inequality is:

$\frac{13}{3} < x < 1.25$

In this range of values for $x$, the inequality is true.