Find the intervals in which ( x − 3 ) ( x 2 − 1 ) > 0 Options: A . &nbsp ( − ∞ , − 1 ) B . &nbsp ( − 1 , 1 ) C . &nbsp ( 1 , 3 ) D . &nbsp ( 3 , ∞ )

Answer: Option B : B and D ( x − 3 ) ( x 2 − 1 ) = ( x − 3 ) ( x − 1 ) ( x + 1 ) L e t P = ( x − 3 ) ( x − 1 ) ( x + 1 ) Find the points on the number line where P=0 P = 0 ⇒ x ϵ { − 1 , 1 , 3 } Split the number line into 4 using these points. ( − ∞ , − 1 ) , ( − 1 , 1 ) , ( 1 , 3 ) & ( 3 , ∞ ) ( − ∞ , − 1 ) : ( x − 3 ) < 0 , ( x − 1 ) < 0 & ( x + 1 ) < 0 ⇒ P < 0 ( − 1 , 1 ) : ( x − 3 ) < 0 , ( x − 1 ) < 0 & ( x + 1 ) > 0 ⇒ P > 0 ( 1 , 3 ) : ( x − 3 ) < 0 , ( x − 1 ) > 0 & ( x + 1 ) > 0 ⇒ P < 0 ( 3 , ∞ ) : ( x − 3 ) > 0 , ( x − 1 ) > 0 & ( x + 1 ) > 0 ⇒ P > 0 ∴ P > 0 , w h e n x ϵ ( − 1 , 1 ) ∪ ( 3 , ∞ )

Find the intervals in which (x−3)(x2−1)>0

Question

Find the intervals in which $(x - 3) (x^{2} - 1) > 0$

Options:

A .

(- \infty, - 1)

B .

(- 1, 1)

C .

(1, 3)

D .

(3, \infty)

Answer: Option B
:
B and D

$(x - 3) (x^{2} - 1) = (x - 3) (x - 1) (x + 1) L e t P = (x - 3) (x - 1) (x + 1)$
Find the points on the number line where P=0
$P = 0 \Rightarrow x ϵ {- 1, 1, 3}$
Split the number line into 4 using these points.
$(- \infty, - 1), (- 1, 1), (1, 3) & (3, \infty) (- \infty, - 1) : (x - 3) < 0, (x - 1) < 0 & (x + 1) < 0 \Rightarrow P < 0 (- 1, 1) : (x - 3) < 0, (x - 1) < 0 & (x + 1) > 0 \Rightarrow P > 0 (1, 3) : (x - 3) < 0, (x - 1) > 0 & (x + 1) > 0 \Rightarrow P < 0 (3, \infty) : (x - 3) > 0, (x - 1) > 0 & (x + 1) > 0 \Rightarrow P > 0$
$∴ P > 0, w h e n x ϵ (- 1, 1) \cup (3, \infty)$