Solve each quadratic inequality. Give the solution set in interval notation.
(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 \(x + 2 = 0\). These give the points \(x = 5\) and \(x = -2\).
2. Use the critical points to divide the number line into three intervals: \(( -\infty, -2 )\), \((-2, 5)\), and \((5, \infty)\).
3. Test a sample value from each interval in the expression \((x - 5)(x + 2)\) to determine whether the product is positive or negative in that interval.
4. For each inequality, determine which intervals satisfy the condition (\(\geq 0\), \(> 0\), \(\leq 0\), or \(< 0\)) and whether to include the critical points based on whether the inequality is strict or not.
5. Write the solution set in interval notation by combining the intervals that satisfy the inequality, including or excluding the endpoints as appropriate.

Key Concepts

Factoring Quadratic Expressions

Factoring involves expressing a quadratic expression as a product of two binomials. In this problem, the quadratic is already factored as (x - 5)(x + 2). Understanding factoring helps identify the roots or zeros of the quadratic, which are critical points for solving inequalities.

Sign Analysis of Quadratic Expressions

Sign analysis determines where a quadratic expression is positive, negative, or zero by testing intervals defined by its roots. Since the expression is factored, the roots x = 5 and x = -2 split the number line into intervals. Evaluating the sign of the product in each interval helps solve inequalities.

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 and brackets for inclusive inequalities. Expressing solutions in interval notation clearly communicates the ranges of x that satisfy the given quadratic inequalities.