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: $2x^2-9x-18\le 0$.
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 $(2x+a)(x+b)\le 0$, 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 $(2x+a)(x+b)\le 0$ 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.