Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. -x^2 + 4x + 1 ≥ 0

Verified step by step guidance

1. Rewrite the inequality in standard quadratic form: \(-x^2 + 4x + 1 \geq 0\).
2. Multiply the entire inequality by -1 to make the quadratic term 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 on the number line by using the roots found in the previous step, and test points from each interval in the inequality \(x^2 - 4x - 1 \leq 0\) to see where the inequality holds true.
5. Write the solution set in interval notation based on the intervals where the inequality is satisfied.

Key Concepts

Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like '≥', '≤', '>', or '<'. In this case, the inequality -x² + 4x + 1 ≥ 0 indicates that we are looking for values of x that make the quadratic expression non-negative. Understanding how to manipulate and solve inequalities is crucial for finding the solution set.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. In this problem, the quadratic -x² + 4x + 1 will be analyzed to determine where it is greater than or equal to zero.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but not 'b'. Writing the solution set of the inequality in interval notation is essential for clearly communicating the range of x values that satisfy the inequality.