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Question

Consider the inequality log(x+3)(x2x)<1. Then

A
x(3,2) if 3<x<2
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B
x(1,0)(1,3) if 3<x<2
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C
x(1,0)(1,3) if 2<x<0 and 1<x<
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D
x(3,2) if 2<x<0 and 1<x<
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Solution

The correct option is C x(1,0)(1,3) if 2<x<0 and 1<x<
For log to be defined,
x+3>0, x+31 and x2x>0
x>3, x2 and x(,0)(1,)
x(3,2)(2,0)(1,)

Now, log(x+3)(x2x)<1
If x+3>1
i.e., x>2
i.e., x(2,0)(1,) (1)
then x2x<x+3
x22x3<0(x3)(x+1)<0
x(1,3) (2)
Hence, from (1) and (2),
x(1,0)(1,3)

If 0<x+3<1
i.e., x(3,2) (3)
then x2x>x+3
x22x3>0(x3)(x+1)>0x(,1)(3,) (4)
Hence, from (3) and (4),
x(3,2)

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