Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x² - 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. Identify 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 determine where the product is negative. For example, test \(x = -3\) in the first interval, \(x = 0\) in the second, and \(x = 4\) in the third.
5. Based on the test results, write the solution set as the interval(s) where the inequality holds true, using interval notation.

Key Concepts

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the values of the variable that make the quadratic expression true for a given inequality sign. This typically requires factoring the quadratic, determining the critical points (roots), and testing intervals between these points to see where the inequality holds.

Factoring Quadratic Expressions

Factoring is the process of expressing a quadratic polynomial as a product of two binomials. For example, x^2 - x - 6 factors into (x - 3)(x + 2). Factoring helps identify the roots of the quadratic, which are essential for analyzing the inequality.

Interval Notation

Interval notation is a way to represent sets of numbers on the number line. It uses parentheses and brackets to indicate open or closed intervals, respectively. For inequalities, interval notation concisely expresses the solution set where the inequality holds true.