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. 3x^2+16x<−5

Verified step by step guidance

1. Rewrite the inequality in standard form: \(3x^2 + 16x + 5 < 0\).
2. Find the roots of the quadratic equation \(3x^2 + 16x + 5 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = 16\), and \(c = 5\).
3. Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
4. Use the roots to divide the number line into intervals and test a point from each interval in the inequality \(3x^2 + 16x + 5 < 0\) to determine where the inequality holds true.
5. Express the solution set in interval notation based on the intervals where the inequality is satisfied.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value using inequality signs (e.g., <, >, ≤, ≥). To solve these inequalities, one typically finds the roots of the corresponding polynomial equation and tests intervals between these roots to determine where the inequality holds true.

Graphing on a Number Line

Graphing the solution set of an inequality on a number line visually represents the values that satisfy the inequality. Solutions are indicated with open or closed circles depending on whether the endpoints are included (open for < or >, closed for ≤ or ≥), and shaded regions show the intervals of solutions.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses to denote open intervals (not including endpoints) and brackets for closed intervals (including endpoints). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5.