Question

(3x-2)(x-3)^3(x+1)^3(x+2)^4<0

Answer 1

Ответ:

Объяснение:

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Answer 2

To solve the inequality (3x-2)(x-3)^3(x+1)^3(x+2)^4 < 0, you can use the following steps:

1. Find the critical values by setting each factor equal to zero:

   • 3x - 2 = 0 => 3x = 2 => x = 2/3
   • (x - 3) = 0 => x = 3
   • (x + 1) = 0 => x = -1
   • (x + 2) = 0 => x = -2

2. Create intervals based on these critical values. The critical values divide the real number line into five intervals: (-∞, -2), (-2, -1), (-1, 2/3), (2/3, 3), and (3, ∞).

3. Test each interval to determine when the expression (3x-2)(x-3)^3(x+1)^3(x+2)^4 is less than 0 (negative). You can choose test points in each interval and see whether the expression is positive or negative at those points.

   • Interval (-∞, -2): Choose x = -3 (a value less than -2): (3(-3) - 2)(-3 - 3)^3(-3 + 1)^3(-3 + 2)^4 = (-9 - 2)(-6)^3(-2)^3(-1)^4 = (-11)(-216)(-8)(1) = 19008 Since it's positive, this interval is not a solution.

   • Interval (-2, -1): Choose x = -1.5 (a value between -2 and -1): (3(-1.5) - 2)(-1.5 - 3)^3(-1.5 + 1)^3(-1.5 + 2)^4 = (-4.5 - 2)(-4.5)^3(-0.5)^3(0.5)^4 = (-6.5)(-91.125)(-0.125)(0.0625) = 0.3623046875 Since it's positive, this interval is not a solution.

   • Interval (-1, 2/3): Choose x = 0 (a value between -1 and 2/3): (3(0) - 2)(0 - 3)^3(0 + 1)^3(0 + 2)^4 = (-2)(-27)(1)(16) = 864 Since it's positive, this interval is not a solution.

   • Interval (2/3, 3): Choose x = 1 (a value between 2/3 and 3): (3(1) - 2)(1 - 3)^3(1 + 1)^3(1 + 2)^4 = (3 - 2)(-2)^3(2)^3(3)^4 = 1(-8)(8)(81) = -5184 Since it's negative, this interval is a solution.

   • Interval (3, ∞): Choose x = 4 (a value greater than 3): (3(4) - 2)(4 - 3)^3(4 + 1)^3(4 + 2)^4 = (12 - 2)(1)^3(5)^3(6)^4 = 10(1)(125)(1296) = 1620000 Since it's positive, this interval is not a solution.

So, the solution to the inequality (3x-2)(x-3)^3(x+1)^3(x+2)^4 < 0 is the interval (2/3, 3).