Solve each quadratic inequality. Give the solution set in interval notation. 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 \(1\), then write \(x^2 + x - 12\) as \((x + 4)(x - 3)\).
4. Set each factor equal to zero to find critical points: \(x + 4 = 0\) gives \(x = -4\), and \(x - 3 = 0\) gives \(x = 3\).
5. Use the critical points to divide the number line into intervals \((-\infty, -4)\), \((-4, 3)\), and \((3, \infty)\), then test a value from each interval in the inequality \((x + 4)(x - 3) < 0\) to determine where the product is negative.

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.

Factoring and Solving Quadratic Equations

To solve quadratic inequalities, first rewrite the inequality in standard form and solve the corresponding quadratic equation by factoring or using the quadratic formula. The roots divide the number line into intervals to test for inequality solutions.

Interval Notation and Test Intervals

After finding critical points from the quadratic equation, use interval notation to express solution sets. Test values from each interval determine where the inequality holds true, allowing you to write the solution as a union of intervals.