Solve each quadratic inequality. Give the solution set in interval notation. x(x-1)≤6

Verified step by step guidance

1. Rewrite the inequality in standard form by moving all terms to one side: \(x(x-1) \leq 6\) becomes \(x^2 - x - 6 \leq 0\).
2. Factor the quadratic expression \(x^2 - x - 6\) by finding two numbers that multiply to \(-6\) and add to \(-1\). This gives \((x - 3)(x + 2) \leq 0\).
3. Identify the critical points by setting each factor equal to zero: \(x - 3 = 0\) gives \(x = 3\), and \(x + 2 = 0\) gives \(x = -2\).
4. Determine the intervals to test based on the critical points: \((-\infty, -2)\), \((-2, 3)\), and \((3, \infty)\). Test a value from each interval in the inequality \((x - 3)(x + 2) \leq 0\) to see where it holds true.
5. Combine the intervals where the inequality is true, including the points where the expression equals zero, and express 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 its graph.

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.

Interval Notation and Test Intervals

After finding critical points, the number line is split 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 valid x-values.