Solve each quadratic inequality. Give the solution set in interval notation. x² - 2 > x

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Rewrite the inequality so that one side is zero by subtracting \( x \) from both sides: \( x^{2} - 2 > x \) becomes \( x^{2} - x - 2 > 0 \).

Factor the quadratic expression \( x^{2} - x - 2 \). Look for two numbers that multiply to \(-2\) and add to \(-1\). This factors as \( (x - 2)(x + 1) \).

Set each factor equal to zero to find the critical points: \( x - 2 = 0 \) gives \( x = 2 \), and \( x + 1 = 0 \) gives \( x = -1 \). These points divide the number line into intervals.

Test a value from each interval in the inequality \( (x - 2)(x + 1) > 0 \) to determine where the product is positive. The intervals to test are \( (-\infty, -1) \), \( (-1, 2) \), and \( (2, \infty) \).

Based on the test results, write the solution set in interval notation, including only the intervals where the inequality holds true (greater than zero). Remember that the inequality is strict, so do not include the critical points themselves.

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주요 개념

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Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than or less than a value or another expression. Solving it requires finding the range of x-values that make the inequality true, often by analyzing the related quadratic equation.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation by setting it equal to zero. This helps find critical points (roots) that divide the number line into intervals to test for the inequality.

Interval Notation and Testing Intervals

After finding the roots, the number line is split into intervals. Test points from each interval in the inequality to determine where it holds true. Express the solution set using interval notation, which concisely represents all valid x-values.

Solve each quadratic inequality. Give the solution set in interval notation. x2 - 2 > x