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

Verified step by step guidance

1. Rewrite the inequality clearly: \(-(x + 1)^2 \geq 0\).
2. Recognize that \((x + 1)^2\) is a perfect square and is always greater than or equal to zero for all real \(x\).
3. Multiply by the negative sign outside the square, which makes \(-(x + 1)^2\) less than or equal to zero, since the square is nonnegative and the negative sign flips the inequality direction.
4. Set the expression equal to zero to find critical points: \(-(x + 1)^2 = 0\) which simplifies to \((x + 1)^2 = 0\).
5. Solve \((x + 1)^2 = 0\) to find \(x = -1\). Use this to determine where the inequality holds and express the solution set in interval notation.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set greater than or less than a value, such as ≥ 0. Solving it requires finding the values of the variable that make the inequality true, often by analyzing the sign of the quadratic expression.

Properties of Squares and Non-Positivity

Since squares of real numbers are always non-negative, expressions like -(x + 1)^2 are always less than or equal to zero. Understanding this helps determine when the inequality holds, especially recognizing when the expression equals zero or is negative.

Interval Notation

Interval notation is a concise way to express solution sets of inequalities using intervals and endpoints. It uses parentheses for open intervals and brackets for closed intervals, indicating whether endpoints are included or excluded.