(x-3)^2*(x+5)*(x+1)^4/(x^2-9)(x+2)>=0

Предыдущий
вопрос Следующий
вопрос

To solve the inequality, we need to find the critical points where the expression is equal to zero and where it is undefined.

The critical points are:
x = -5, -1, 3, -2, 3

Therefore, the intervals we need to test are (-∞,-5), (-5,-3), (-3,-2), (-2,-1), (-1,3), (3,∞).

We will test each interval by choosing a test point within each interval and plugging it into the expression to determine the sign of the expression in that interval.

For x = -6: (-5-6)^2(-6+5)(-6+1)^4/(-6^2-9)(-6+2) = (-1)^2(-1)(-5)^4/(36-9)(-4) = 100/1350 > 0
For x = -4: (-5-4)^2(-4+5)(-4+1)^4/(-4^2-9)(-4+2) = (-9)^2(1)(-3)^4/(16-9)(-2) = -4374 > 0
For x = - 2.5: (-5-2.5)^2(-2.5+5)(-2.5+1)^4/(-2.5^2-9)(-2.5+2) = (-7.5)^2(2.5)(-1.5)^4/(6.25-9)(-0.5) = 14.4 > 0
For x = -2: (-3)^2(-2+5)(-2+1)^4/(-2^2-9)(-2+2) = (3)^2(3)(-1)^4/(4-9)(0) = 0
For x = -1.5: (-3)^2(-1.5+5)(-1.5+1)^4/(-1.5^2-9)(-1.5+2) = (2.25)(3.5)(-0.5)^4/(2.25-9)(0.5) = 0
For x = 0: (-3)^2(0+5)(0+1)^4/(0^2-9)(0+2) = (3)^2(5)(1)^4/(-9)(2) = -5
For x = 2: (-3)^2(2+5)(2+1)^4/(2^2-9)(2+2) = (1)^2(7)(3)^4/(4-9)(4) = 63
For x = 4: (-3)^2(4+5)(4+1)^4/(4^2-9)(4+2) = (1)^2(9)(5)^4/7(6) = 1125/7 > 0
For x = 5: (-3)^2(5+5)(5+1)^4/(5^2-9)(5+2) = (2)^2(10)(6)^4/(25-9)(7) = 7200/224 > 0

Therefore, the solution to the inequality is x ∈ (-∞,-5) U (-1,3).

I hope this helps! Let me know if you have any questions.