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+3x>0

Verified step by step guidance

1. Step 1: Set the inequality to zero to find the critical points. Solve the equation 2x^2 + 3x = 0.
2. Step 2: Factor the quadratic equation. Notice that you can factor out an x, giving x(2x + 3) = 0.
3. Step 3: Solve for x by setting each factor equal to zero. This gives x = 0 and 2x + 3 = 0.
4. Step 4: Solve the equation 2x + 3 = 0 for x. Subtract 3 from both sides to get 2x = -3, then divide by 2 to find x = -3/2.
5. Step 5: Use the critical points x = 0 and x = -3/2 to test intervals on the number line. Choose test points in the intervals (-∞, -3/2), (-3/2, 0), and (0, ∞) to determine where the inequality 2x^2 + 3x > 0 holds true.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value using inequality symbols (>, <, ≥, ≤). To solve these inequalities, one typically finds the roots of the polynomial, determines the intervals on the number line, and tests these intervals to see where the inequality holds true.

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 intervals) or excluded (open intervals). 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 where the inequality is satisfied is marked, 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.