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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1:29 minutes

Problem 29

Textbook Question

Solve each inequality. Give the solution set in interval notation. See Example 4. -5<5+2x<11

Verified step by step guidance

Start by isolating the variable term in the middle of the compound inequality: subtract 5 from all three parts of the inequality: \(-5 - 5 < 5 + 2x - 5 < 11 - 5\).

Simplify the inequality: \(-10 < 2x < 6\).

Divide each part of the inequality by 2 to solve for \(x\): \(\frac{-10}{2} < \frac{2x}{2} < \frac{6}{2}\).

Simplify the inequality: \(-5 < x < 3\).

Express the solution set in interval notation: \((-5, 3)\).

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Key Concepts

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

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They can be represented using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving inequalities involves finding the range of values that satisfy the given condition.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses parentheses and brackets to indicate whether the endpoints are included in the set. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints. This notation is essential for clearly expressing the solution set of inequalities.

Compound Inequalities

Compound inequalities involve two or more inequalities that are combined into one statement, often using the word 'and' or 'or'. In the given question, -5 < 5 + 2x < 11 is a compound inequality that requires solving for x in a way that satisfies both conditions simultaneously. Understanding how to manipulate and solve these inequalities is crucial for finding the correct solution set.