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. First, rewrite the inequality so that one side is zero. Add 5 to both sides to get: \$3x^{2} + 16x + 5 < 0$.
2. Next, factor the quadratic expression \$3x^{2} + 16x + 5\(. To do this, look for two numbers that multiply to \)3 \times 5 = 15$ and add to 16.
3. Once you find the factors, express the quadratic as a product of two binomials: \((ax + b)(cx + d) < 0\).
4. Find the critical points by setting each factor equal to zero and solving for \(x\). These points divide the number line into intervals.
5. Test a value from each interval in the original inequality to determine where the inequality holds true. Then, express the solution set in interval notation and graph it on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to 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 and Solving Quadratic Equations

To solve polynomial inequalities, especially quadratics, it is essential to rewrite the inequality in standard form and factor the polynomial if possible. Factoring helps find the roots (zeros) of the polynomial, which divide the number line into intervals to test for the inequality.

Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality holds, solutions are expressed using interval notation, which concisely represents all values satisfying the inequality. Graphing on a number line visually shows these solution intervals, using open or closed circles to indicate whether endpoints are included.