Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. -(x + 1)² ≥ 0

Verified step by step guidance

1. Rewrite the inequality to clearly see the expression: \(-(x + 1)^2 \geq 0\).
2. Recognize that \((x + 1)^2\) is a square term, which is always greater than or equal to zero for all real \(x\).
3. Since the expression is \(-(x + 1)^2\), it is the negative of a square, so it is always less than or equal to zero.
4. Set the expression equal to zero to find critical points: \(-(x + 1)^2 = 0\) which simplifies to \((x + 1)^2 = 0\).
5. Solve for \(x\) from \((x + 1)^2 = 0\) to find \(x = -1\). Use this to determine where the inequality holds and express the solution set in interval notation.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic function is compared to zero or another value using inequality symbols (>, <, ≥, ≤). Solving them requires finding the set of x-values that make the inequality true, often by analyzing the sign of the quadratic expression.

Properties of Quadratic Functions

A quadratic function is a polynomial of degree two, typically in the form ax² + bx + c. Its graph is a parabola, which opens upward if a > 0 and downward if a < 0. Understanding the vertex and the shape helps determine where the function is positive, negative, or zero.

Interval Notation

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. It uses parentheses () for open intervals (excluding endpoints) and brackets [] for closed intervals (including endpoints), helping clearly express where the inequality holds true.