Question

Решите,пожалуйста 1)(x+4)(x+2)(x-0)<0 2)(x+4)(x+3)x<0 3)(2x²-16x)(4x+4)>0 4)(x²-25)(x²-9)<0

Answer 1

1)(x+4)(x+2)(x-0)<0
x=-4  x=-2  x=0
             _                      +                    _                    +
-----------------(-4)--------------(-2)-----------------(0)----------------------
x∈(-∞;-4) U (-2;0)
2)(x+4)(x+3)x<0
x=-4  x=-3  x=0
             _                      +                    _                    +
-----------------(-4)--------------(-3)-----------------(0)----------------------
x∈(-∞;-4) U (-3;0)
3)(2x²-16x)(4x+4)>0
2x(x-8)(4x+4)>0
x=0  x=8  x=-1
             _                      +                    _                    +
-----------------(-1)--------------(0)-----------------(8)----------------------
x∈(-1;0) U (8;∞)
4)(x²-25)(x²-9)<0
(x-5)(x+5)(x-3)(x+3)<0
x=5  x=-5  x=3  x=-3
           +                  _                      +                    _                    +
-----------------(-5)--------------(-3)-----------------(3)------------(5)----------
x∈(-5;-3) U (3;5)

Answer 2

Решение во вложении...........

Answer 3

1) (x+4)(x+2)(x-0) < 0

To solve this inequality, we can use the concept of interval notation. We need to find the intervals where the expression (x+4)(x+2)(x-0) is less than zero.

First, let's find the critical points by setting each factor equal to zero and solving for x:

x + 4 = 0 x = -4

x + 2 = 0 x = -2

x - 0 = 0 x = 0

Now, we can create a number line and mark these critical points on it:

-4   -2   0

Next, we need to test the sign of the expression (x+4)(x+2)(x-0) in each interval:

In the interval (-∞, -4), we can choose x = -5: (-5 + 4)(-5 + 2)(-5 - 0) = (-1)(-3)(-5) = 15 Since the expression is positive in this interval, it does not satisfy the inequality.

In the interval (-4, -2), we can choose x = -3: (-3 + 4)(-3 + 2)(-3 - 0) = (1)(-1)(-3) = 3 The expression is negative in this interval and satisfies the inequality.

In the interval (-2, 0), we can choose x = -1: (-1 + 4)(-1 + 2)(-1 - 0) = (3)(1)(-1) = -3 The expression is negative in this interval and satisfies the inequality.

In the interval (0, ∞), we can choose x = 1: (1 + 4)(1 + 2)(1 - 0) = (5)(3)(1) = 15 Since the expression is positive in this interval, it does not satisfy the inequality.

Therefore, the solution to the inequality (x+4)(x+2)(x-0) < 0 is: -4 < x < -2 or 0 < x

2) (x+4)(x+3)x < 0

To solve this inequality, we can use a similar approach. We need to find the intervals where the expression (x+4)(x+3)x is less than zero.

Let's find the critical points by setting each factor equal to zero and solving for x:

x + 4 = 0 x = -4

x + 3 = 0 x = -3

x = 0

Now, let's create a number line and mark these critical points on it:

-4   -3   0

Next, we need to test the sign of the expression (x+4)(x+3)x in each interval:

In the interval (-∞, -4), we can choose x = -5: (-5 + 4)(-5 + 3)(-5) = (-1)(-2)(-5) = -10 The expression is negative in this interval and satisfies the inequality.

In the interval (-4, -3), we can choose x = -3.5: (-3.5 + 4)(-3.5 + 3)(-3.5) = (0.5)(-0.5)(-3.5) = 0.875 The expression is positive in this interval, so it does not satisfy the inequality.

In the interval (-3, 0), we can choose x = -2: (-2 + 4)(-2 + 3)(-2) = (2)(1)(-2) = -4 The expression is negative in this interval and satisfies the inequality.

In the interval (0, ∞), we can choose x = 1: (1 + 4)(1 + 3)(1) = (5)(4)(1) = 20 The expression is positive in this interval, so it does not satisfy the inequality.

Therefore, the solution to the inequality (x+4)(x+3)x < 0 is: -4 < x < -3 or 0 < x

3) (2x²-16x)(4x+4) > 0

To solve this inequality, we can again find the critical points by setting each factor equal to zero and solving for x:

2x² - 16x = 0 2x(x - 8) = 0

x = 0 or x = 8

Now we can create a number line and mark these critical points on it:

0    8

Next, we need to test the sign of the expression (2x²-16x)(4x+4) in each interval:

In the interval (-∞, 0), we can choose x = -1: (2(-1)² - 16(-1))(4(-1) + 4) = (2 + 16)(-4) = -72 The expression is negative in this interval and does not satisfy the inequality.

In the interval (0, 8), we can choose x = 1: (2(1)² - 16(1))(4(1) + 4) = (2 - 16)(8) = -112 The expression is negative in this interval and does not satisfy the inequality.

In the interval (8, ∞), we can choose x = 9: (2(9)² - 16(9))(4(9) + 4) = (162 - 144)(40) = 720 The expression is positive in this interval and satisfies the inequality.

Therefore, the solution to the inequality (2x²-16x)(4x+4) > 0 is: 8 < x

4) (x²-25)(x²-9) < 0

To solve this inequality, we can find the critical points by setting each factor equal to zero and solving for x:

x² - 25 = 0 (x - 5)(x + 5) = 0

x = -5 or x = 5

x² - 9 = 0 (x - 3)(x + 3) = 0

x = -3 or x = 3

Now we can create a number line and mark these critical points on it:

-5   -3   3   5

Next, we need to test the sign of the expression (x²-25)(x²-9) in each interval:

In the interval (-∞, -5), we can choose x = -6: ((-6)² - 25)((-6)² - 9) = (36 - 25)(36 - 9) = (11)(27) = 297 The expression is positive in this interval and does not satisfy the inequality.

In the interval (-5, -3), we can choose x = -4: ((-4)² - 25)((-4)² - 9) = (16 - 25)(16 - 9) = (-9)(7) = -63 The expression is negative in this interval and satisfies the inequality.

In the interval (-3, 3), we can choose x = 0: ((0)² - 25)((0)² - 9) = (-25)(-9) = 225 The expression is positive in this interval and does not satisfy the inequality.

In the interval (3, 5), we can choose x = 4: ((4)² - 25)((4)² - 9) = (16 - 25)(16 - 9) = (-9)(7) = -63 The expression is negative in this interval and satisfies the inequality.

In the interval (5, ∞), we can choose x = 6: ((6)² - 25)((6)² - 9) = (36 - 25)(36 - 9) = (11)(27) = 297 The expression is positive in this interval and does not satisfy the inequality.

Therefore, the solution to the inequality (x²-25)(x²-9) < 0 is: -5 < x < -3 or 3 < x < 5