Question

(X^2+5x+6)*x*(x-3)^3 / (x^2+6x+9)*(x-5) <= 0

Answer 1

Сначала решим квадратные уравнения в неравенстве (я это сделала под цифрами 1 и 2), под ниже, под первым неравенством я записала преобразованное, нашла корни (х), при которых неравенство будет равно 0, и дальше методом интервалов: взяла любой корень, подставила, выяснила знак и прочередовала (надеюсь, вам это уде обьясняли).
Надеюсь, помогла)

Answer 2

Let's break down the problem by finding the critical points and testing intervals.

Critical points:

• x = 0 (from the x term)
• x = 3 (from the (x-3) factor)
• x = 5 (from the (x-5) factor)
• x = -2 or x = -3 (from the x^2+6x+9 = (x+3)^2 factor)

Testing intervals:

1. x < -3 Plugging in a value less than -3 into the inequality gives: positive * negative * negative / negative * negative <= 0 This is true since there are an odd number of negative factors.

2. -3 < x < -2 Plugging in a value between -3 and -2 into the inequality gives: positive * negative * negative / positive * negative <= 0 This is false since there are an even number of negative factors.

3. -2 < x < 0 Plugging in a value between -2 and 0 into the inequality gives: positive * negative * positive / positive * negative <= 0 This is true since there are an odd number of negative factors.

4. 0 < x < 3 Plugging in a value between 0 and 3 into the inequality gives: positive * positive * negative / positive * negative <= 0 This is false since there are an even number of negative factors.

5. 3 < x < 5 Plugging in a value between 3 and 5 into the inequality gives: positive * positive * negative / positive * negative <= 0 This is true since there are an odd number of negative factors.

6. x > 5 Plugging in a value greater than 5 into the inequality gives: positive * positive * positive / positive * positive <= 0 This is false since there are no negative factors.

Therefore, the solution to the inequality is: x < -3 or -2 < x < 0 or 3 < x < 5.