Question

(x-2) (x-4) (x-9)>0​

Answer 1

Ответ:

__________

__________

Answer 2

To solve the inequality (x-2)(x-4)(x-9) > 0, we can use the method of sign charts.

Step 1: Find the critical points by setting each factor equal to zero and solving for x:

x - 2 = 0 --> x = 2 x - 4 = 0 --> x = 4 x - 9 = 0 --> x = 9

Step 2: Plot the critical points on a number line:

luaCopy code
       2          4          9
       |-----|-----|-----|

Step 3: Test the intervals between the critical points:

For the interval (-∞, 2), we can choose x = 0 as a test point: (0 - 2)(0 - 4)(0 - 9) = (-2)(-4)(-9) = -72, which is negative.

For the interval (2, 4), we can choose x = 3 as a test point: (3 - 2)(3 - 4)(3 - 9) = (1)(-1)(-6) = 6, which is positive.

For the interval (4, 9), we can choose x = 6 as a test point: (6 - 2)(6 - 4)(6 - 9) = (4)(2)(-3) = -24, which is negative.

For the interval (9, ∞), we can choose x = 10 as a test point: (10 - 2)(10 - 4)(10 - 9) = (8)(6)(1) = 48, which is positive.

Step 4: Determine the sign of the expression in each interval:

luaCopy code
       2          4          9
       |-----|-----|-----|
         -     +     -     +

Step 5: Write the solution based on the sign chart:

The inequality (x-2)(x-4)(x-9) > 0 is satisfied when x is in the intervals (-∞, 2) and (4, 9).

Therefore, the solution to the inequality is x ∈ (-∞, 2) ∪ (4, 9).