Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x(x+1)<12

Verified step by step guidance

1. Rewrite the inequality in standard form by moving all terms to one side: \(x(x+1) - 12 < 0\).
2. Expand the left side: \(x^2 + x - 12 < 0\).
3. Factor the quadratic expression: find two numbers that multiply to \(-12\) and add to \$1\(, so factor as \)(x + 4)(x - 3) < 0$.
4. Determine the critical points by setting each factor equal to zero: \(x + 4 = 0\) gives \(x = -4\), and \(x - 3 = 0\) gives \(x = 3\).
5. Use a sign chart or test values in the intervals \((-\infty, -4)\), \((-4, 3)\), and \((3, \infty)\) to find where the product \((x + 4)(x - 3)\) is less than zero, then 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 sign of the quadratic expression over intervals determined by its roots.

Factoring and Finding Roots

To solve quadratic inequalities, first rewrite the inequality in standard form and factor the quadratic expression if possible. The roots (solutions to the corresponding quadratic equation) divide the number line into intervals where the expression's sign can be tested.

Interval Notation and Sign Testing

After finding the roots, use interval notation to express the solution set. Test points from each interval to determine where the inequality holds true. This method helps identify which intervals satisfy the inequality and should be included in the solution.