Solve each quadratic inequality. Give the solution set in interval notation. x²-x-6>0

Verified step by step guidance

1. Rewrite the inequality in standard quadratic form: \(x^2 - x - 6 > 0\).
2. Factor the quadratic expression on the left side: \(x^2 - x - 6 = (x - 3)(x + 2)\).
3. Determine the critical points by setting each factor equal to zero: \(x - 3 = 0\) gives \(x = 3\), and \(x + 2 = 0\) gives \(x = -2\).
4. Use the critical points to divide the number line into three intervals: \((-\infty, -2)\), \((-2, 3)\), and \((3, \infty)\).
5. Test a value from each interval in the inequality \((x - 3)(x + 2) > 0\) to determine where the product is positive, then write the solution set in interval notation based on these results.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than or less than zero (or another value). Solving it means finding all x-values that make the inequality true, often by analyzing the sign of the quadratic expression over different intervals.

Factoring Quadratic Expressions

Factoring rewrites a quadratic expression as a product of two binomials. For example, x² - x - 6 factors to (x - 3)(x + 2). Factoring helps identify the roots, which divide the number line into intervals to test for the inequality.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as ranges of values. After finding roots, the number line is split into intervals where the quadratic is positive or negative. Testing points in each interval determines where the inequality holds, and the solution is written using interval notation.