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 a value from each interval in the inequality \(x^2 - 2x - 6 < 0\) to see where the inequality holds true.
5. Write the solution set as the interval(s) where the inequality is satisfied, using interval notation.

Key Concepts

Solving Quadratic Inequalities

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

Factoring and Finding Roots of Quadratic Equations

To solve quadratic inequalities, it is essential to find the roots of the quadratic equation by factoring, completing the square, or using the quadratic formula. These roots divide the number line into intervals that can be tested to determine where the inequality is satisfied.

Interval Notation

Interval notation is a concise way to represent the set of all solutions to an inequality. It uses parentheses and brackets to indicate open or closed intervals, showing where the variable values lie on the number line that satisfy the inequality.