Solve each quadratic inequality. Give the solution set in interval notation. x²+4x>-1

Verified step by step guidance

1. Rewrite the inequality in standard form by moving all terms to one side: \(x^{2} + 4x > -1\) becomes \(x^{2} + 4x + 1 > 0\).
2. Find the roots of the corresponding quadratic equation \(x^{2} + 4x + 1 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=4\), and \(c=1\).
3. Calculate the discriminant \(\Delta = b^{2} - 4ac\) to determine the nature of the roots and then find the exact roots.
4. Use the roots to divide the number line into intervals. Test a value from each interval in the inequality \(x^{2} + 4x + 1 > 0\) to determine where the inequality holds true.
5. Write the solution set in interval notation based on the intervals where the inequality is satisfied.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than or less than a value, such as x² + 4x > -1. Solving it requires finding the range of x-values that make the inequality true, often by analyzing the related quadratic equation.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation (e.g., x² + 4x + 1 = 0) to find critical points. These points divide the number line into intervals to test for the inequality's truth.

Interval Notation and Testing Intervals

After finding critical points, use interval notation to express solution sets. Test values from each interval in the original inequality to determine where it holds true, then write the solution as intervals showing all valid x-values.