(x^2-6x+8)(x^2-4)/x^3-8>=0

Предыдущий
вопрос Следующий
вопрос

We first factor the left side of the inequality:

(x^2 - 6x + 8)(x^2 - 4) = (x - 2)(x - 4)(x + 2)(x - 2) = (x - 2)^2(x - 4)(x + 2)

Now, we have the inequality:

(x - 2)^2(x - 4)(x + 2)/(x^3 - 8) >= 0

To solve this inequality, we first find the critical points by setting the numerator and denominator equal to zero:

Numerator: x = 2, x = 4
Denominator: x = 2

Now, we can test the intervals created by these critical points:

Interval 1: (-∞, 2)
Interval 2: (2, 4)
Interval 3: (4, ∞)

Now, we test a point in each interval to determine the sign of the expression:

For x = 0 (in Interval 1): (-) (+) (-) = (-)
For x = 3 (in Interval 2): (+) (+) (-) = (-)
For x = 5 (in Interval 3): (+) (-) (+) = (-)

Since we are looking for where the expression is greater than or equal to 0, the solution set is:

x ∈ (2, 4] and x ≠ 2