Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 2x^2+x<15

Verified step by step guidance

1. Rewrite the inequality in standard form: \(2x^2 + x - 15 < 0\).
2. Find the roots of the equation \(2x^2 + x - 15 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = 1\), and \(c = -15\).
3. Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
4. Solve for the roots using the quadratic formula, which will give you the critical points.
5. Use the critical points to test intervals on the number line to determine where the inequality \(2x^2 + x - 15 < 0\) holds true, and express the solution set in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value using inequality symbols (e.g., <, >, ≤, ≥). To solve these inequalities, one typically rearranges the expression to one side, setting it to zero, and then determines the intervals where the polynomial is either positive or negative.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. 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.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the solutions to an inequality. Each interval derived from the inequality is marked on the line, using open or closed circles to indicate whether the endpoints are included. This graphical representation helps in understanding the range of values that satisfy the inequality.