$\frac{x^{2}-2}{1-2 x}>1$

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Precalculus

Evaluate rational exponents

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\frac{x^{2}-2}{1-2 x}>1

Rewrite Inequality: Rewrite the inequality to have zero on one side. $\newline$ $(\frac{x^2 - 2}{1 - 2x}) - 1 > 0$
Combine Terms: Combine terms over a common denominator. $(x^2 - 2 - (1 - 2x))/(1 - 2x) > 0$
Simplify Numerator: Simplify the numerator. $\newline$ $(x^2 - 2 - 1 + 2x)/(1 - 2x) > 0$ $\newline$ $(x^2 + 2x - 3)/(1 - 2x) > 0$
Factor Numerator: Factor the numerator. $\newline$ $(x + 3)(x - 1)/(1 - 2x) > 0$
Rewrite Denominator: Notice that $(1 - 2x)$ is the same as $- (2x - 1)$ , so rewrite the denominator. $(x + 3)(x - 1)/-(2x - 1) > 0$
Multiply and Flip Inequality: Multiply both sides by $-1$ to get rid of the negative sign and flip the inequality. $\newline$ $-\frac{(x + 3)(x - 1)}{2x - 1} < 0$
Find Zeros and Undefined: Find the zeros and undefined points of the expression. $\newline$ Zeros: $x = -3$ , $x = 1$ $\newline$ Undefined: $x = \frac{1}{2}$
Plot Zeros and Test Intervals: Plot the zeros and undefined point on a number line and test intervals. $\newline$ Test intervals: $-\infty, -3$ , $-3, \frac{1}{2}$ , $\frac{1}{2}, 1$ , $1, \infty$
Choose Test Points: Choose test points: $-4$ , $0$ , rac{3}{4}, $2$ and evaluate the expression. $\newline$ Test point $-4$ : -ig((-4 + 3)(-4 - 1)ig)/ig(2(-4) - 1ig) < 0 (True) $\newline$ Test point $0$ : -ig((0 + 3)(0 - 1)ig)/ig(2(0) - 1ig) < 0 (False) $\newline$ Test point rac{3}{4}: -igg(igg(rac{3}{4}igg) + 3igg)igg(igg(rac{3}{4}igg) - 1igg)/igg(2igg(rac{3}{4}igg) - 1igg) < 0 (False) $\newline$ Test point $2$ : $0$ $1$ (True)
Determine Solution Set: Determine the solution set from the number line. $\newline$ Solution set: $(-\infty, -3) \cup (1, \infty)$