Question

Решите неравенства х(х+2)(х-5)≤0

Answer 1

Метод интервалов:
x=0         x= -2        x=5
    -                +                 -                  +
-------- -2 ----------- 0 ------------- 5 -------------
\\\\\\\\\\                       \\\\\\\\\\\\\\\\

x< -2                         -    -    -  |   - 
-2<x<0     x=-1          -   +    -  |  +
0<x<5      x=1           +  +    -  |   -
x>5          x=6           +  +   +  |  +

x∈(-∞; -2]U[0; 5]

Ответ: (-∞; -2]U[0; 5].

Answer 2

Solving the Inequality:

To solve the inequality x(x+2)(x-5) ≤ 0, we can use the concept of sign charts.

1. Find the critical points by setting each factor equal to zero and solving for x: - x = 0 (from x = 0) - x + 2 = 0 (from x = -2) - x - 5 = 0 (from x = 5)

2. Plot these critical points on a number line:

-2 0 5

3. Choose test points in each interval and determine the sign of the expression x(x+2)(x-5):

- Test x = -3: (-3)(-3+2)(-3-5) = (-3)(-1)(-8) = 24 > 0 (positive) - Test x = 1: (1)(1+2)(1-5) = (1)(3)(-4) = -12 < 0 (negative) - Test x = 6: (6)(6+2)(6-5) = (6)(8)(1) = 48 > 0 (positive)

4. Determine the sign of the expression for each interval:

- From -∞ to -2: The expression is positive (+) - From -2 to 0: The expression is negative (-) - From 0 to 5: The expression is positive (+) - From 5 to +∞: The expression is negative (-)

5. Write the solution as an inequality:

x(x+2)(x-5) ≤ 0 is true when -2 ≤ x ≤ 0 or 5 ≤ x.

Therefore, the solution to the inequality is -2 ≤ x ≤ 0 or x ≥ 5.

Solution:

The solution to the inequality x(x+2)(x-5) ≤ 0 is -2 ≤ x ≤ 0 or x ≥ 5.