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Find all possible values of expressions `(2+x^2)/(4-x^2)`

Text Solution

Verified by Experts

The correct Answer is:
`(-oo, -1)cup[(1)/(2),oo)`
`(2+x^(2))/(4-x^(2))=(6-(4-x^(2)))/(4-x^(2))=(6)/(4-x^(2))=1`
Now, `x^(2) ge 0`
` :. -x^(2) le 0`
`implies 4-x^(2) le 4`
`implies (4-x^(2))/(6) le (2)/(3)`
`implies -oo lt (4-x^(2))/(6) lt 0 " or "0 lt(4-x^(2))/(6) le (2)/(3)`
`implies 0 gt (6)/(4-x^(2)) gt -oo gt (4-x^(2))/(6) ge (3)/(2)`
`implies (6)/(4-x^(2)) in (-oo,0) cup [(3)/(2),oo)`
`implies ((6)/(4-x^(2))-1) in (-oo,-1) cup [(1)/(2),oo)`
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