Solve the inequality. $$ 2 x-1-x^{2}

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Solve the inequality. $$ 2 x-1-x^{2}<0 $$

UpStudy AI · Accepted Answer

1. Start with the inequality: $$ 2x - 1 - x^2 < 0 $$ 2. Rewrite the inequality in standard quadratic form: $$ -x^2 + 2x - 1 < 0 $$ 3. Multiply both sides by $$-1$$ (remembering to reverse the inequality sign): $$ x^2 - 2x + 1 > 0 $$ 4. Recognize that the quadratic is a perfect square: $$ (x - 1)^2 > 0 $$ 5. The square $$(x - 1)^2$$ is zero when $$x = 1$$ and positive for all other values of $$x$$. Since we have a strict inequality, the solution is: $$ x \neq 1 $$ 6. In interval notation, the solution is: $$ (-\infty, 1) \cup (1, \infty) $$ The solution to the inequality $$2x - 1 - x^2 < 0$$ is all real numbers except $$x = 1$$. In interval notation, this is $$(-\infty, 1) \cup (1, \infty)$$.

Solve the inequality.

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