Solve each quadratic inequality. Give the solution set in interval notation. x²≤9

Verified step by step guidance

1. Rewrite the inequality \(x^{2} \leq 9\) in a form that makes it easier to analyze by subtracting 9 from both sides: \(x^{2} - 9 \leq 0\).
2. Factor the left-hand side expression using the difference of squares formula: \(x^{2} - 9 = (x - 3)(x + 3)\), so the inequality becomes \((x - 3)(x + 3) \leq 0\).
3. Identify the critical points by setting each factor equal to zero: \(x - 3 = 0\) gives \(x = 3\), and \(x + 3 = 0\) gives \(x = -3\). These points divide the number line into intervals to test.
4. Test values from each interval determined by the critical points \((-\infty, -3)\), \((-3, 3)\), and \((3, \infty)\) in the inequality \((x - 3)(x + 3) \leq 0\) to determine where the product is less than or equal to zero.
5. Based on the test results, write the solution set in interval notation, including the points where the expression equals zero since the inequality is 'less than or equal to'.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it means finding all x-values that satisfy the inequality, often by analyzing the related quadratic equation and testing intervals.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation (e.g., x² = 9) to find critical points. These points divide the number line into intervals to test for inequality satisfaction.

Interval Notation

Interval notation expresses the solution set of inequalities using brackets and parentheses to indicate inclusive or exclusive bounds. For example, [−3, 3] represents all x between −3 and 3, including the endpoints.