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

Verified step by step guidance

1. Start by expanding the expression \(-(x + 1)^2\). This becomes \(-(x^2 + 2x + 1)\).
2. Simplify the expression to \(-x^2 - 2x - 1\).
3. Set the inequality \(-x^2 - 2x - 1 \geq 0\).
4. Notice that the expression \(-(x + 1)^2\) is always less than or equal to zero because it is a negative square. Therefore, the inequality holds when \(-(x + 1)^2 = 0\).
5. Solve \(-(x + 1)^2 = 0\) by setting \(x + 1 = 0\), which gives \(x = -1\). The solution set in interval notation is \([-1, -1]\).

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically using symbols like '≥', '≤', '>', or '<'. To solve these inequalities, one must determine the values of the variable that satisfy the inequality, often by finding the roots of the corresponding quadratic equation and testing intervals between these roots.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval [a, b] includes both a and b, while (a, b) does not include them. This notation is essential for expressing the solution set of inequalities succinctly.

Graphing Quadratics

Graphing quadratics involves plotting the quadratic function on a coordinate plane to visualize its shape, which is a parabola. The vertex, axis of symmetry, and intercepts are key features that help in understanding the behavior of the quadratic. This graphical representation aids in determining where the quadratic is above or below a certain value, which is crucial for solving inequalities.