Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2-x-6>0

Verified step by step guidance

1. Start by rewriting the inequality: \(x^2 - x - 6 > 0\).
2. Factor the quadratic expression on the left side. Find two numbers that multiply to \(-6\) and add to \(-1\). This gives the factorization: \((x - 3)(x + 2) > 0\).
3. Determine the critical points by setting each factor equal to zero: \(x - 3 = 0\) gives \(x = 3\), and \(x + 2 = 0\) gives \(x = -2\). These points divide the number line into three intervals: \((-\infty, -2)\), \((-2, 3)\), and \((3, \infty)\).
4. Test a value from each interval in the inequality \((x - 3)(x + 2) > 0\) to see where the product is positive. For example, test \(x = -3\) in \((-\infty, -2)\), \(x = 0\) in \((-2, 3)\), and \(x = 4\) in \((3, \infty)\).
5. Based on the test results, write the solution set in interval notation, including only the intervals where the inequality holds true (product is greater than zero).

Key Concepts

Quadratic Inequalities

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

Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic expression as a product of two binomials. For example, x^2 - x - 6 factors to (x - 3)(x + 2). Factoring helps identify the roots, which are critical points for determining where the quadratic changes sign.

Interval Notation and Sign Analysis

Interval notation expresses solution sets as intervals on the number line. After finding roots, sign analysis tests values in intervals between roots to determine where the inequality holds true. This method helps write the solution set accurately in interval notation.