Solve each quadratic inequality. Give the solution set in interval notation. See Example 1. (a) (x - 5)(x + 2) ≥ 0 (b) (x - 5)(x + 2) > 0 (c) (x - 5)(x + 2) ≤ 0 (d) (x - 5)(x + 2) < 0

Verified step by step guidance

1. Identify the critical points by setting each factor equal to zero: solve \(x - 5 = 0\) and \(x + 2 = 0\) to find \(x = 5\) and \(x = -2\). These points divide the number line into intervals.
2. Determine the intervals to test based on the critical points: \((-\infty, -2)\), \((-2, 5)\), and \((5, \infty)\).
3. Choose a test point from each interval and substitute it into the expression \((x - 5)(x + 2)\) to check whether the product is positive, negative, or zero in that interval.
4. For each inequality, decide which intervals satisfy the condition: for \(\geq 0\) and \(> 0\), select intervals where the product is positive (include zeros for \(\geq 0\)); for \(\leq 0\) and \(< 0\), select intervals where the product is negative (include zeros for \(\leq 0\)).
5. Write the solution sets in interval notation, including or excluding the critical points based on whether the inequality is strict (\(>\) or \(<\)) or inclusive (\(\geq\) or \(\leq\)).

Key Concepts

Factoring and Zero-Product Property

Factoring expresses a quadratic expression as a product of linear factors, such as (x - 5)(x + 2). The zero-product property states that if a product equals zero, then at least one factor must be zero. This helps identify critical points (x = 5 and x = -2) where the expression changes sign.

Sign Analysis of Quadratic Expressions

Sign analysis involves determining where a quadratic expression is positive, negative, or zero by testing intervals defined by its roots. By evaluating the sign of each factor in intervals around the critical points, you can find where the product meets the inequality conditions (≥ 0, > 0, ≤ 0, < 0).

Interval Notation for Solution Sets

Interval notation is a concise way to represent sets of real numbers that satisfy inequalities. It uses parentheses for strict inequalities (excludes endpoints) and brackets for inclusive inequalities (includes endpoints). This notation clearly communicates the solution intervals for the quadratic inequalities.