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. x²+5x+4>0

Verified step by step guidance

1. Start by rewriting the inequality: \(x^2 + 5x + 4 > 0\).
2. Factor the quadratic expression on the left side: \(x^2 + 5x + 4 = (x + 1)(x + 4)\).
3. Identify the critical points by setting each factor equal to zero: \(x + 1 = 0\) gives \(x = -1\), and \(x + 4 = 0\) gives \(x = -4\).
4. Use the critical points to divide the real number line into intervals: \((-\infty, -4)\), \((-4, -1)\), and \((-1, \infty)\).
5. Test a value from each interval in the inequality \((x + 1)(x + 4) > 0\) 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 is the process of expressing a quadratic polynomial as a product of two binomials. For example, x² + 5x + 4 factors to (x + 1)(x + 4). Factoring helps identify the roots of the polynomial, which are critical points for determining where the polynomial changes sign.

Interval Notation and Number Line Graphing

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial is positive or negative. Using critical points from factoring, intervals are tested to determine where the inequality holds true.