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Question

The complete set of values of x that satisfies the expression 1log4(x+1x+2)>1log4(x+3) is

A
(1,)
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B
(,2)
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C
(3,2)
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D
(3,1)
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Solution

The correct option is C (3,2)
1log4(x+1x+2)>1log4(x+3) (1)
log is defined when
x+3>0x(3,) (i)
And x+1x+2>0
x(,2)(1,) (ii)
From (i) and (ii)
x(3,2)(1,) (2)

Case 1: x(1,) ...(3)
x+1x+2(0,1) and x+3(2,)
log4(x+1x+2)<0 and log4(x+3)>0
which does not satisfies inequation (1)

Case 2: x(3,2) ...(4)
x+1x+2(2,) and x+3(0,1)
log4(x+1x+2)>0 and log4(x+3)<0
log4(x+1x+2)>log4(x+3)
x+1x+2>x+3
x+1x+2x+3>0
(x2+4x+5)x+2>0
(x+2)2+1x+2<0
x<2 ...(5)

Now, from (2),(4) and (5)
x(3,2)

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