Making assumptions about where parentheses should be,
√(6 + x - 2x^2) < 2 - x
√ yields non-negative numbers so 2 - x must be greater than equal to 0. Therefore, we can square both sides of the inequality and leave the inequality symbol as is.
(√(6 + x - 2x^2))^2 < (2 - x)^2
6 + x - 2x^2 < (2 - x)^2
(3 + 2x)(2 - x) < (2 - x)^2
(3 + 2x)(2 - x) - (2 - x)^2 < 0
(3 + 2x - (2 - x))(2 - x) < 0
(1 + 3x)(2 - x) < 0
We've already determined 2 - x ≥ 0. Therefore,
1 + 3x < 0
x < -1/3
The expression within the √ must be non-negative. Therefore,
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Verified answer
Making assumptions about where parentheses should be,
√(6 + x - 2x^2) < 2 - x
√ yields non-negative numbers so 2 - x must be greater than equal to 0. Therefore, we can square both sides of the inequality and leave the inequality symbol as is.
(√(6 + x - 2x^2))^2 < (2 - x)^2
6 + x - 2x^2 < (2 - x)^2
(3 + 2x)(2 - x) < (2 - x)^2
(3 + 2x)(2 - x) - (2 - x)^2 < 0
(3 + 2x - (2 - x))(2 - x) < 0
(1 + 3x)(2 - x) < 0
We've already determined 2 - x ≥ 0. Therefore,
1 + 3x < 0
x < -1/3
The expression within the √ must be non-negative. Therefore,
(3 + 2x)(2 - x) > 0
We've already determined 2 - x ≥ 0. Therefore,
3 + 2x ≥ 0
x ≥ -3/2
Therefore,
x < -1/3 and x ≥ -3/2
-3/2 ≤ x < -1/3