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Question

Find the range of x for the following expression:

x23x+2x2+3x+21


A

(0,81)

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B

(,1)

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C

(,2)(2,1)(1,0)

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D

None of these

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Solution

The correct option is C

(,2)(2,1)(1,0)


x23x+2=(x1)(x2)and x2+3x+2=(x+1)(x+2)

For x<2:
x23x+2x2+3x+2x0
So the interval is (,2)

For 2<x<1, we obtain:
x23x+2(x2+3x+2)
2(x2+2)0
xR
So, the interval is (2,1)

For 1<x<1, we obtain:
x23x+2x2+3x+2
3x3x
x0
So the interval is (1,0]

For 1x2, we obtain:
(x23x+2)x2+3x+2
2(x2+2)0
So, no value is possible in this interval.

For 2<x:
x23x+2x2+3x+2
3x3x
x0
So, no value is possible in this interval.

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