Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x(x-1)≤6

Verified step by step guidance

1. Rewrite the inequality in standard form by moving all terms to one side: \(x(x-1) - 6 \leq 0\).
2. Expand the expression: \(x^2 - x - 6 \leq 0\).
3. Factor the quadratic expression: \((x - 3)(x + 2) \leq 0\).
4. Determine the critical points by setting each factor equal to zero: \(x - 3 = 0\) gives \(x = 3\), and \(x + 2 = 0\) gives \(x = -2\).
5. Test the intervals determined by the critical points \((-\infty, -2)\), \((-2, 3)\), and \((3, \infty)\) to see where the product \((x - 3)(x + 2)\) is less than or equal to zero, then write the solution set in interval notation.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it requires finding the range of 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 by setting the expression equal to the boundary value. This helps identify critical points that divide the number line into intervals for testing the inequality.

Interval Notation and Testing Intervals

After finding critical points, the number line is divided into intervals. Each interval is tested to determine if it satisfies the inequality. The solution set is then expressed in interval notation, which concisely represents all x-values that make the inequality true.