Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 2x^2-9x≤18

Verified step by step guidance

1. First, rewrite the inequality in standard form by moving all terms to one side: \(2x^2 - 9x - 18 \leq 0\).
2. Next, factor the quadratic expression \(2x^2 - 9x - 18\). Look for two numbers that multiply to \(-36\) (the product of \(2\) and \(-18\)) and add to \(-9\).
3. Once factored, the expression should look like \((2x + a)(x + b) \leq 0\), where \(a\) and \(b\) are the numbers found in the previous step.
4. Determine the critical points by setting each factor equal to zero: \(2x + a = 0\) and \(x + b = 0\). Solve these equations to find the values of \(x\).
5. Use the critical points to test intervals on the number line. Choose test points from each interval to determine where the inequality \((2x + a)(x + b) \leq 0\) holds true, and express the solution set in interval notation.

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value using inequality symbols (e.g., ≤, ≥, <, >). To solve these inequalities, one typically finds the roots of the corresponding quadratic equation and then tests intervals to determine where the inequality holds true.

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 interval) or excluded (open interval). For example, (a, b) represents all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. This is essential for solving quadratic equations and inequalities, as it allows one to find the roots easily. For example, the quadratic expression ax^2 + bx + c can often be factored into (px + q)(rx + s), where p, q, r, and s are constants.