Find the complete range of values of x that satisfies |x+3|>x2−11x+30.
Find the complete range of values of x that satisfies |x+3|>x2−11x+30.
The correct option is D (3, 9)
Conventional Approach :
Given inequality is |x+3|>x2−11x+30
We consider 2 cases
Case (i) : x>−3
|x+3|=x+3
x+3>x2−11x+30⇒x2−12x+27<0
(x−9)(x−3)<0
x= (3,9)
Case (ii) ; x<−3
|x+3|= -x-3
−x−3>x2−11x+30
⇒x2−10x+33<0
⇒x2−10x+25+8<0=(x−5)2+8<0
(x−5)2+8 is always a positive quantity, irrespective of the value of x. Hence we do not have any value for x in the interval. Hence the valid interval for x is (3,9).
Shortcut- Elimination Approach
Step 1: Choose a value for x which is present in 2 answer options and absent in the other 2. Eg) x=4 satisfies the range specified in option (a) and (d), but does not form a part of the range in option (b) and (c)
At x=4
LHS= |4+3|=7
RHS= 16 - 44+30 = 2. LHS>RHS. It is satisfied. This means that answer options, which do not include x=4 can be directly eliminated as they can NEVER be the right answer
Step 2- Choose a value for x which is there in one of the remaining 2 options and absent in the other
Eg) Take x=10. This option satisfies the range in option (a) but is absent in option (d)
At x=10
LHS= |10+3|=13
RHS= 100-110+30= 20
LHS<RHS. This violates the condition in the question. This means that option (a) can never be the right answer.
Thus, by elimination, answer is option (d)
(3, 9)
Conventional Approach :
Given inequality is |x+3|>x2−11x+30
We consider 2 cases
Case (i) : x>−3
|x+3|=x+3
x+3>x2−11x+30⇒x2−12x+27<0
(x−9)(x−3)<0
x= (3,9)
Case (ii) ; x<−3
|x+3|= -x-3
−x−3>x2−11x+30
⇒x2−10x+33<0
⇒x2−10x+25+8<0=(x−5)2+8<0
(x−5)2+8 is always a positive quantity, irrespective of the value of x. Hence we do not have any value for x in the interval. Hence the valid interval for x is (3,9).
Shortcut- Elimination Approach
Step 1: Choose a value for x which is present in 2 answer options and absent in the other 2. Eg) x=4 satisfies the range specified in option (a) and (d), but does not form a part of the range in option (b) and (c)
At x=4
LHS= |4+3|=7
RHS= 16 - 44+30 = 2. LHS>RHS. It is satisfied. This means that answer options, which do not include x=4 can be directly eliminated as they can NEVER be the right answer
Step 2- Choose a value for x which is there in one of the remaining 2 options and absent in the other
Eg) Take x=10. This option satisfies the range in option (a) but is absent in option (d)
At x=10
LHS= |10+3|=13
RHS= 100-110+30= 20
LHS<RHS. This violates the condition in the question. This means that option (a) can never be the right answer.
Thus, by elimination, answer is option (d)
