Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x²>16

Verified step by step guidance

1. Rewrite the inequality \(x^2 > 16\) by bringing all terms to one side: \(x^2 - 16 > 0\).
2. Recognize that \(x^2 - 16\) is a difference of squares, which factors as \((x - 4)(x + 4) > 0\).
3. Determine the critical points by setting each factor equal to zero: \(x - 4 = 0\) gives \(x = 4\), and \(x + 4 = 0\) gives \(x = -4\).
4. Use the critical points to divide the number line into three intervals: \((-\infty, -4)\), \((-4, 4)\), and \((4, \infty)\). Test a value from each interval in the inequality \((x - 4)(x + 4) > 0\) to determine where the product is positive.
5. Based on the test results, write the solution set in interval notation, including only the intervals where the inequality holds true (remember the inequality is strict, so do not include the points where the expression equals zero).

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than or less than a value, such as x² > 16. Solving it requires finding the values of x that make the inequality true, often by analyzing the related quadratic equation and its graph.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation (e.g., x² = 16) to find critical points. These points divide the number line into intervals that can be tested to determine where the inequality holds.

Interval Notation and Number Line Testing

After finding critical points, use interval notation to express solution sets clearly. Test values from each interval on the number line to check if they satisfy the inequality, then write the solution as a union of intervals where the inequality is true.