Question

(y+6)(y-3)<(-2y-2)(y+7)-2(y+1) ​

Answer 1

Ответ:

y^2-3y+6y-18<-2y^2-14y-2y-14-2y-2

-2y^2-y^2-14y-2y-2y+3y-6y<-2-14+18

-3y^2-21y<2

-3y(y+7)<2

-3y<2 или у+7<2

у>2/3; у<-5

Надеюсь, что правильно сделал

Answer 2

Let's solve the given inequality step by step:

Given inequality: (y + 6)(y - 3) < (-2y - 2)(y + 7) - 2(y + 1)

Expand the expressions on both sides:

(y^2 - 3y + 6y - 18) < (-2y^2 - 14y - 2y - 14) - 2y - 2

Simplify the expressions:

y^2 + 3y - 18 < -2y^2 - 18y - 16

Move all terms to one side of the inequality:

y^2 + 2y^2 + 3y + 18y - 18 + 16 < 0

Combine like terms:

3y^2 + 21y - 2 < 0

Now we need to solve this quadratic inequality. To do so, we'll find the critical points by setting the expression equal to zero and then determine the sign of the expression within the intervals formed by these critical points.

Factor the quadratic expression:

3y^2 + 21y - 2 = 0

This equation doesn't factor easily, so let's use the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 3, b = 21, and c = -2:

y = (-21 ± √(21^2 - 4 * 3 * -2)) / (2 * 3) y = (-21 ± √(441 + 24)) / 6 y = (-21 ± √465) / 6

The solutions are:

y = (-21 + √465) / 6 y = (-21 - √465) / 6

Now, let's analyze the intervals between these critical points and test the sign of the expression within those intervals to determine the solution range.

Interval 1: y < (-21 - √465) / 6 Interval 2: (-21 - √465) / 6 < y < (-21 + √465) / 6 Interval 3: y > (-21 + √465) / 6

Pick a test point from each interval and substitute it into the expression:

Test point for interval 1: y = -10 3(-10)^2 + 21(-10) - 2 = -280, which is negative

Test point for interval 2: y = 0 3(0)^2 + 21(0) - 2 = -2, which is negative

Test point for interval 3: y = 10 3(10)^2 + 21(10) - 2 = 328, which is positive

Based on these tests, we can see that the expression is negative in intervals 1 and 2, and positive in interval 3.

Therefore, the solution to the inequality 3y^2 + 21y - 2 < 0 is: (-21 - √465) / 6 < y < (-21 + √465) / 6.