please help ....
solve the inequations:-
Answers
Answer:
Step-by-step explanation:
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Answer:
⇒ x ∈ (-3 , 2) ∪ (4 , ∞)
Step-by-step explanation:
Given :
To solve for range of x, if
x³ - 3x² - 10x + 24 > 0
Solution :
⇒ x³ - 3x² - 10x + 24 > 0
⇒ x³ - 2x² - x² + 2x - 12x + 24 > 0
⇒ x² (x - 2) - x (x - 2) - 12 (x - 2) > 0
⇒ (x - 2) (x² - x - 12) > 0
⇒ (x - 2) (x² - 4x + 3x - 12) > 0
⇒ (x - 2) [ x( x - 4) + 3(x - 4) ] > 0
⇒ (x - 2) (x + 3) (x - 4) > 0
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-∞ -3 0 2 4 ∞
If x ∈ (-∞ , -3)
x - 2 < 0 (x - 2) must be negative.
x + 3 < 0 (x + 3) must be negative.
x - 4 < 0 (x - 4) must be negative.
(x - 2) ( x + 3) (x - 4) < 0 ( - × - × - = -),
Hence, x ∉ (-∞ , -3) (Because, The product must be greater than 0, not less than 0.)
If x = - 3,.
⇒ (x - 2) ( x + 3) (x - 4) = (-3 - 2) ( -3 + 3) (-3 - 4) = (-5) (0) (-7) = 0,.
⇒ x ≠ -3 (Because, The product must be greater than 0, not 0)
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If x ∈ (-3 , 2)
x - 2 < 0 must be negative.
x + 3 > 0 must be positive.
x - 4 < 0 must be negative.
(x - 2) ( x + 3) (x - 4) > 0 ( - × + × - = +)
⇒ ∴ x ∈ (-3 , 2) (Because, The product must be greater than 0)
If x = 2,
(x - 2) ( x + 3) (x - 4) = (2 - 2) ( 2 + 3) (2 - 4) = (0) (5) ( -2) = 0
⇒ x ≠ 2 (Because, The product must be greater than 0, not 0)
If x ∈ (2 , 4)
x - 2 > 0 must be positive.
x - 4 < 0 must be negative.
x + 3 > 0 must be positive.
(x - 2) (x + 3) (x - 4) < 0 ( + × - × + = -)
⇒ x ∉ (2 , 4) (Because, The product must be greater than 0, not less than 0.)
If x = 4,
(x - 2) ( x + 3) (x - 4) = (4 - 2) (4 + 3) (4 - 4) = (2) (7) (0) = 0
⇒ x ≠ 4 (Because, The product must be greater than 0, not 0)
If x ∈ (4 , ∞)
x - 2 > 0 (x - 2) must be positive
x - 4 > 0 (x - 2) must be positive
x + 3 > 0 (x - 2) must be positive
(x - 2) ( x + 3) (x - 4) < 0 ( + × + × + = +)
⇒ ∴ x ∈ (4 , ∞) (Because, The product must be greater than 0.)
⇒ x ∈ (-3 , 2) ∪ (4 , ∞)