Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. -x² + 4x + 1 ≥ 0

Verified step by step guidance

1. Rewrite the inequality in a standard form by moving all terms to one side: \(-x^2 + 4x + 1 \geq 0\).
2. Multiply the entire inequality by \(-1\) to make the quadratic coefficient positive, remembering to reverse the inequality sign: \(x^2 - 4x - 1 \leq 0\).
3. Find the roots of the 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 \(c=-1\).
4. Determine the intervals defined by the roots and test values within each interval to see where the inequality \(x^2 - 4x - 1 \leq 0\) holds true.
5. Write the solution set in interval notation based on the intervals where the inequality is satisfied.

Key Concepts

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the values of the variable that make the inequality true. This typically requires rewriting the inequality in standard form, finding the roots of the corresponding quadratic equation, and determining the intervals where the quadratic expression is positive or negative.

Factoring and Finding Roots

To solve the inequality, first find the roots of the quadratic by factoring or using the quadratic formula. These roots divide the number line into intervals. Testing points in each interval helps determine where the inequality holds true.

Interval Notation

Interval notation is a concise way to express the solution set of inequalities. It uses parentheses and brackets to indicate open or closed intervals, showing where the variable satisfies the inequality. For example, [a, b] includes endpoints, while (a, b) excludes them.