Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Linear Inequalities

College Algebra

1. Equations & Inequalities

Linear Inequalities

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1:53 minutes

Problem 16

Textbook Question

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

Rewrite the inequality in standard form: \(3x^2 + 16x + 5 < 0\).

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 \(c = 5\).

Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.

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.

Express the solution set in interval notation based on the intervals where the inequality is satisfied.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.