Solve each quadratic inequality. Give the solution set in interval notation. x²>16

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Rewrite the inequality \(x^{2} > 16\) to isolate zero on one side: \(x^{2} - 16 > 0\).

Recognize that \(x^{2} - 16\) is a difference of squares, which factors as \((x - 4)(x + 4) > 0\).

Determine the critical points by setting each factor equal to zero: \(x - 4 = 0\) gives \(x = 4\), and \(x + 4 = 0\) gives \(x = -4\).

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 see where the product is positive.

Based on the test results, write the solution set in interval notation, including only the intervals where the inequality holds true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 all x-values that make the inequality true, often by analyzing the related quadratic equation.

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 to test for the inequality's truth.

Interval Notation and Number Line Testing

After finding critical points, use interval notation to express solution sets. Test values from each interval in the inequality to determine where it holds true, then write the solution as unions of intervals.

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