Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. (a) -(x + 1)(x + 2) ≥ 0 (b) -(x + 1)(x + 2) > 0 (c) -(x + 1)(x + 2) ≤ 0 (d) -(x + 1)(x + 2) < 0

Verified step by step guidance

1. Step 1: Identify the quadratic expression and set it equal to zero to find the critical points. For the inequality \(- (x + 1)(x + 2)\), set \((x + 1)(x + 2) = 0\) to find the critical points.
2. Step 2: Solve \((x + 1)(x + 2) = 0\) to find the critical points. The solutions are \(x = -1\) and \(x = -2\). These points divide the number line into intervals.
3. Step 3: Determine the sign of the expression \(- (x + 1)(x + 2)\) in each interval created by the critical points \(-\infty, -2, -1, \infty\). Choose test points from each interval to determine the sign.
4. Step 4: For each inequality, determine which intervals satisfy the inequality. For example, for \(- (x + 1)(x + 2) \geq 0\), find intervals where the expression is non-negative.
5. Step 5: Write the solution set for each inequality in interval notation, considering whether the endpoints are included (\([\) or \(]\)) or not (\((\) or \()) based on the inequality sign.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions of the form ax^2 + bx + c > 0, < 0, ≥ 0, or ≤ 0. To solve these inequalities, one must first find the roots of the corresponding quadratic equation, which helps determine the intervals to test for the inequality. The solution set is then expressed in interval notation, indicating the ranges of x that satisfy the inequality.

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, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Test Points

Test points are specific values chosen from the intervals created by the roots of the quadratic inequality. By substituting these points into the inequality, one can determine whether the entire interval satisfies the inequality. This method is essential for identifying which intervals are part of the solution set.