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.

Verified step by step guidance

1. Start by writing the inequality: \(x^2 + 2x < 0\).
2. Factor the left-hand side expression: \(x^2 + 2x = x(x + 2)\).
3. Set the product less than zero: \(x(x + 2) < 0\). This means the product of two factors is negative.
4. Determine the critical points by setting each factor equal to zero: \(x = 0\) and \(x + 2 = 0 \Rightarrow x = -2\). These points divide the number line into intervals.
5. Test values from each interval \((-\infty, -2)\), \((-2, 0)\), and \((0, \infty)\) in the inequality \(x(x + 2) < 0\) to find where the product is negative, then express the solution set in interval notation.

Key Concepts

Solving Polynomial Inequalities

Solving polynomial inequalities involves finding the values of the variable that make the inequality true. This typically requires factoring the polynomial, determining critical points where the expression equals zero, and testing intervals between these points to see where the inequality holds.

Factoring Quadratic Expressions

Factoring quadratic expressions means rewriting the quadratic as a product of two binomials. For example, x² + 2x can be factored as x(x + 2). Factoring helps identify the roots or zeros of the polynomial, which are essential for analyzing the inequality.

Interval Notation and Number Line Graphing

Interval notation is a way to represent solution sets using intervals, such as (a, b) or [a, b), indicating which numbers satisfy the inequality. Graphing on a number line visually shows these intervals, helping to understand where the polynomial is positive or negative.