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Question

If the inequality loga(x2x2)>loga(x2+2x+3) is known to be satisfied for x=94 and if the solution set of the inequality is (p,q), then value of 2(p+q) is


Solution

loga(x2x2)>loga(x2+2x+3)
For log to be defined
x2x2>0
(x2)(x+1)>0
x(,1)(2,)   (i)
And 
x2+2x+3>0
x22x3<0
(x+1)(x3)<0
x(1,3)   (ii)
From (i) and (ii) 
x(2,3)   (iii)

Now, at x=94 given expression is satisfied. Substituting this in given inequality, we get 
loga(1316)>loga(3916)
which is possible when 0<a<1
Given,
loga(x2x2)>loga(x2+2x+3)
x2x2<x2+2x+3
2x23x5<0
(x+1)(2x5)<0
x(1,52)   (iv)

Therefore, from (iii) and (iv) we get,
x(2,52)
(p,q)(2,52)
2(p+q)=2(2+52)=9
 

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