Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. -x² + 2x + 6 > 0

Verified step by step guidance

1. Rewrite the inequality in a standard form by moving all terms to one side: \(-x^{2} + 2x + 6 > 0\).
2. Multiply the entire inequality by \(-1\) to make the quadratic coefficient positive, remembering to reverse the inequality sign: \(x^{2} - 2x - 6 < 0\).
3. Find the roots of the quadratic equation \(x^{2} - 2x - 6 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=-2\), and \(c=-6\).
4. Determine the intervals defined by the roots and test values within these intervals to see where the inequality \(x^{2} - 2x - 6 < 0\) holds true.
5. Write the solution set in interval notation based on the intervals where the inequality is satisfied.

Key Concepts

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the values of the variable that make the quadratic expression greater than or less than zero. This typically requires rewriting the inequality in standard form, finding the roots of the corresponding quadratic equation, and testing intervals between the roots to determine where the inequality holds true.

Factoring and Finding Roots

To solve the inequality, you first find the roots of the quadratic by factoring or using the quadratic formula. These roots divide the number line into intervals. Knowing the roots helps identify critical points where the expression changes sign, which is essential for testing the solution intervals.

Interval Notation

Interval notation is a concise way to express the solution set of inequalities. It uses parentheses and brackets to indicate open or closed intervals, representing all values that satisfy the inequality. Understanding how to write and interpret interval notation is crucial for clearly communicating the solution.