Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. x² - 2 > x

Verified step by step guidance

1. Rewrite the inequality in standard form by moving all terms to one side: \(x^2 - 2 > x\) becomes \(x^2 - x - 2 > 0\).
2. Factor the quadratic expression \(x^2 - x - 2\) by finding two numbers that multiply to \(-2\) and add to \(-1\). This gives \((x - 2)(x + 1) > 0\).
3. Determine the critical points by setting each factor equal to zero: \(x - 2 = 0\) gives \(x = 2\), and \(x + 1 = 0\) gives \(x = -1\). These points divide the number line into intervals.
4. Test a value from each interval \((-\infty, -1)\), \((-1, 2)\), and \((2, \infty)\) in the inequality \((x - 2)(x + 1) > 0\) to see where the product is positive.
5. Based on the test results, write the solution set in interval notation including only the intervals where the inequality holds true.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than or less than a value, such as x² - 2 > x. Solving it requires finding the values of x that make the inequality true, often by analyzing the related quadratic equation.

Solving Quadratic Equations

To solve a quadratic inequality, first rewrite it as a quadratic equation by setting it equal to zero (e.g., x² - x - 2 > 0 becomes x² - x - 2 = 0). Finding the roots of this equation helps identify critical points that divide the number line into intervals for testing.

Interval Notation and Testing Intervals

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