(х-1)(х-4)(х-8)(х-16)<0

Предыдущий
вопрос Следующий
вопрос

To solve this inequality, we first need to find the critical points where the expression on the left side of the inequality changes sign.

The critical points occur when each factor is equal to zero:
1) x - 1 = 0 --> x = 1
2) x - 4 = 0 --> x = 4
3) x - 8 = 0 --> x = 8
4) x - 16 = 0 --> x = 16

Now, we need to test the intervals created by these critical points to determine when the expression is less than zero. We can do this by using a number line:

---|---o---o---o---o---|---

We test the intervals:
1) (-∞, 1): Choose x = 0
(0-1)(0-4)(0-8)(0-16) = (-1)(-4)(-8)(-16) = -512 < 0 (True)

2) (1, 4): Choose x = 3
(3-1)(3-4)(3-8)(3-16) = (2)(-1)(-5)(-13) = 130 > 0 (False)

3) (4, 8): Choose x = 6
(6-1)(6-4)(6-8)(6-16) = (5)(2)(-2)(-10) = 200 > 0 (False)

4) (8, 16): Choose x = 10
(10-1)(10-4)(10-8)(10-16) = (9)(6)(2)(-6) = -648 < 0 (True)

5) (16, ∞): Choose x = 20
(20-1)(20-4)(20-8)(20-16) = (19)(16)(12)(4) = 14496 > 0 (False)

The solution to the inequality (х-1)(х-4)(х-8)(х-16) < 0 is:
x ∈ (1, 4) U (8, 16)