Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x²+4x>-1

Verified step by step guidance

1. Rewrite the inequality to have zero on one side by adding 1 to both sides: \(x^2 + 4x + 1 > 0\).
2. Identify the quadratic expression: \(x^2 + 4x + 1\). To analyze the inequality, find the roots of the corresponding quadratic equation \(x^2 + 4x + 1 = 0\) 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 \(\Delta = b^2 - 4ac = 4^2 - 4(1)(1)\) to determine the nature of the roots.
4. Use the roots found 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. Express the solution set as an interval or union of intervals based on the test results, using interval notation.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than, less 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 it equal to zero. This helps find critical points (roots) that divide the number line into intervals to test for the inequality's truth.

Interval Notation and Testing Intervals

After finding the roots, 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 x-values that make the inequality true.