Question

(x-3)(x-4)^2(х-10)<0

Answer 1

Методом интервалов
точки 3,4,10
(-бесконечности;3)(3;4)(4;10)

Answer 2

To solve this inequality, we need to find the intervals on the x-axis where the expression is negative. We can do this by considering the signs of each factor and using the sign rules for multiplication.

First, let's look at each factor:

• (x-3) is negative when x < 3, positive when x > 3, and zero when x = 3.
• (x-4)^2 is positive for all x, and zero when x = 4.
• (x-10) is negative when x > 10, positive when x < 10, and zero when x = 10.

Next, let's consider the sign of the entire expression:

• If all three factors are negative, then the expression is negative.
• If two factors are negative and one factor is positive, then the expression is negative.
• If one factor is negative and two factors are positive, then the expression is positive.
• If all three factors are positive, then the expression is positive.

Using these rules, we can construct a sign chart:

scssCopy code
         x < 3      3 < x < 4      4 < x < 10      x > 10
(x-3)    -          -              +              +
(x-4)^2  +          0              +              +
(x-10)   -          -              -              +

From the sign chart, we can see that the expression is negative in the intervals (3, 4) and (10, infinity). Therefore, the solution to the inequality is:

Copy code
3 < x < 4 or x > 10

In interval notation, this is:

scssCopy code
(3, 4) U (10, infinity)