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Solve, `(1-sqrt(21-4x-x^(2)))/(x+1) ge 0`. |
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Answer» Correct Answer - `x in [-2-2sqrt(6),-1) uu [-2+2sqrt(6),3]` We have `(1-sqrt(21-4x-x^(2)))/(x+1)ge0` We must have `21-4x-x^(2) ge 0` `implies x^(2)+4x-21 le 0` `implies (x+7)(x-3) le 0` `implies x in [-7,3]` Case I : `x+1 gt 0 implies x gt -1` .......`(i)` So given inequality reduces to `1 ge sqrt(21-4x-x^(2))` `implies 1 ge 21-4x-x^(2)` `implies x^(2)+4x-20 ge 0` `implies (x+2)^(2)-(2sqrt(6))^(2) ge 0` `implies (x+2+2sqrt(6))(x+2-2sqrt(6)) ge 0` `implies x in [-2+2sqrt(6),3]` ............`(ii)` Case II : `x lt -1` So given inequality reduces to `1-sqrt(21-4x-x^(2)) le 0` `implies 1 le sqrt(21-4x-x^(2))` `implies 1 le 21-4x-x^(2)` `implies x^(2)+4x-20 le 0` `implies x in [-2-2sqrt(6),-1)` ...........`(iii)` So from `(ii)` and `(iii)`, `x in [-2-2sqrt(6),-1)uu[-2+2sqrt(6),3]` |
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