Tìm x: ( x - 1 ) . ( x - 4 ) ≥ 0

To solve the inequality (x-1)(x-4) >= 0, we need to find the values of x that make the expression greater than or equal to zero.

First, let's find the critical points by setting the expression equal to zero:
(x-1)(x-4) = 0
x-1 = 0 or x-4 = 0
x = 1 or x = 4

Now, we can create a number line and test the intervals between the critical points and beyond:

Interval 1: (-∞, 1)
Choose a test point, such as x = 0:
(0-1)(0-4) = (-1)(-4) = 4 > 0
Since the expression is positive in this interval, it satisfies the inequality.

Interval 2: (1, 4)
Choose a test point, such as x = 2:
(2-1)(2-4) = (1)(-2) = -2 < 0
Since the expression is negative in this interval, it does not satisfy the inequality.

Interval 3: (4, ∞)
Choose a test point, such as x = 5:
(5-1)(5-4) = (4)(1) = 4 > 0
Since the expression is positive in this interval, it satisfies the inequality.

Therefore, the solution to (x-1)(x-4) >= 0 is x ∈ (-∞, 1] ∪ [4, ∞).

Now, let's solve the inequality (8-x)(2-x) <= 0:

First, let's find the critical points by setting the expression equal to zero:
(8-x)(2-x) = 0
8-x = 0 or 2-x = 0
x = 8 or x = 2

Now, we can create a number line and test the intervals between the critical points and beyond:

Interval 1: (-∞, 2)
Choose a test point, such as x = 0:
(8-0)(2-0) = (8)(2) = 16 > 0
Since the expression is positive in this interval, it does not satisfy the inequality.

Interval 2: (2, 8)
Choose a test point, such as x = 5:
(8-5)(2-5) = (3)(-3) = -9 < 0
Since the expression is negative in this interval, it satisfies the inequality.

Interval 3: (8, ∞)
Choose a test point, such as x = 9:
(8-9)(2-9) = (-1)(-7) = 7 > 0
Since the expression is positive in this interval, it does not satisfy the inequality.

Therefore, the solution to (8-x)(2-x) <= 0 is x ∈ (2, 8].