(Х-2,5)^2·(3х-14)^5<0

Предыдущий
вопрос Следующий
вопрос

To find the solution to this inequality, we first need to factor each part of the inequality and then identify the critical points where the expression equals zero.

Given:

(x-2.5)^2 * (3x-14)^5 < 0

The critical points for the first factor (x-2.5)^2 are x = 2.5 and x = 2.5

The critical points for the second factor (3x-14)^5 are x = 14/3 and x = 14/3

Now, plot these critical points on a number line:

....2.5....14/3......2.5.....

Choose a test point in each of the intervals: (-∞, 2.5), (2.5, 14/3), (14/3, ∞)

Test point in (-∞, 2.5): x = 0
(0-2.5)^2 (3(0)-14)^5 = (2.5)^2 (-14)^5 = 6.25 * (-537824) = -33614

Test point in (2.5, 14/3): x = 3
(3-2.5)^2 (3(3)-14)^5 = (0.5)^2 (9-14)^5 = 0.25 * (-5)^5 = -3125

Test point in (14/3, ∞): x = 4
(4-2.5)^2 (3(4)-14)^5 = (1.5)^2 (12-14)^5 = 2.25 * (-2)^5 = -72

Since the inequality is less than zero, we are looking for the interval where the expression is negative.

From the test points, we can see that the solution is the interval (2.5, 14/3).

Therefore, the solution to the inequality is 2.5 < x < 14/3.