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. Start by rewriting the inequality: \$2x^{2} + 3x > 0$.
2. Factor the left-hand side expression: \(x(2x + 3) > 0\).
3. Identify the critical points by setting each factor equal to zero: \(x = 0\) and \$2x + 3 = 0 \Rightarrow x = -\frac{3}{2}$.
4. Use the critical points to divide the real number line into intervals: \((-\infty, -\frac{3}{2})\), \((-\frac{3}{2}, 0)\), and \((0, \infty)\).
5. Test a value from each interval in the factored inequality \(x(2x + 3) > 0\) to determine where the product is positive, then express the solution set in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols like >, <, ≥, or ≤. Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Factoring Quadratic Polynomials

Factoring quadratic polynomials means rewriting the expression as a product of two binomials or simpler polynomials. This step is crucial for solving inequalities because it helps identify the roots or zeros, which divide the number line into intervals to test for the inequality.

Interval Notation and Graphing on the Number Line

Interval notation is a concise way to represent sets of real numbers that satisfy inequalities, using parentheses and brackets to indicate open or closed intervals. Graphing the solution on a number line visually shows where the polynomial inequality holds true, aiding in understanding and verification.