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, including endpoints if the inequality is non-strict (\(\geq\) or \(\leq\)).

Key Concepts

Solving Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to zero or another value using inequality signs. To solve them, first rewrite the inequality in standard form, then find the roots of the corresponding quadratic equation. These roots divide the number line into intervals to test for where the inequality holds true.

Factoring and Finding Roots of Quadratic Equations

Finding the roots of a quadratic equation involves setting the quadratic expression equal to zero and solving for the variable. This can be done by factoring, completing the square, or using the quadratic formula. The roots are critical points that help determine the intervals for testing the inequality.

Interval Notation and Solution Sets

Interval notation is a concise way to represent sets of real numbers that satisfy inequalities. It uses parentheses and brackets to indicate open or closed intervals, respectively. After determining where the quadratic inequality holds, express the solution set using interval notation to clearly communicate the range of valid values.