Question

A) -2 < -2x + 5 < 3 б) (3-3x)(x-2)(x+7)<0

Answer 1

Ответ:

лови

Объяснение:

Answer 2

To solve these inequalities, we'll work on each of them separately:

A) -2 < -2x + 5 < 3

First, let's isolate -2x in the middle part of the inequality:

-2 < -2x + 5 < 3

Subtract 5 from all parts of the inequality:

-2 - 5 < -2x < 3 - 5

-7 < -2x < -2

Now, divide all parts by -2. Remember that when you multiply or divide by a negative number, you must reverse the direction of the inequality signs:

(-7)/(-2) > (-2x)/(-2) > (-2)/(-2)

7/2 > x > 1

So, the solution for the inequality -2 < -2x + 5 < 3 is 1 < x < 7/2.

B) (3-3x)(x-2)(x+7) < 0

To solve this inequality, we'll use the concept of sign changes at the roots of each factor. First, we find the roots by setting each factor equal to zero:

1. 3 - 3x = 0 => 3x = 3 => x = 1
2. x - 2 = 0 => x = 2
3. x + 7 = 0 => x = -7

Now, we create a number line and mark these roots:

---o---o---o---

Now, test a point in each interval created by these roots. You can use any value in each interval to determine the sign of the expression. Let's test a value to the left of -7 (e.g., -8):

For x < -7: (3-3x)(x-2)(x+7) = (positive)(negative)(negative) = positive

Now, test a value between -7 and 1 (e.g., 0):

For -7 < x < 1: (3-3x)(x-2)(x+7) = (positive)(negative)(positive) = negative

Finally, test a value greater than 1 (e.g., 3):

For x > 1: (3-3x)(x-2)(x+7) = (negative)(positive)(positive) = negative

So, the solution to the inequality (3-3x)(x-2)(x+7) < 0 is x < -7, -7 < x < 1, and x > 2.