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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2:03 minutes

Problem 28

Textbook Question

Solve each inequality. Give the solution set in interval notation. | 3x - 4 | < 2

Verified step by step guidance

Start by understanding the inequality \(|3x - 4| < 2\). This means the expression inside the absolute value, \(3x - 4\), must be between \(-2\) and \(2\).

Set up two separate inequalities to remove the absolute value: \(-2 < 3x - 4 < 2\).

Solve the left inequality: \(-2 < 3x - 4\). Add 4 to both sides to get \(2 < 3x\). Then, divide both sides by 3 to isolate \(x\), resulting in \(\frac{2}{3} < x\).

Solve the right inequality: \(3x - 4 < 2\). Add 4 to both sides to get \(3x < 6\). Then, divide both sides by 3 to isolate \(x\), resulting in \(x < 2\).

Combine the solutions from both inequalities to find the solution set: \(\frac{2}{3} < x < 2\). In interval notation, this is \((\frac{2}{3}, 2)\).

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Key Concepts

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

Absolute Value Inequalities

Absolute value inequalities involve expressions that measure the distance of a number from zero on the number line. The inequality |A| < B means that the expression A lies within the interval (-B, B). Understanding how to interpret and solve these inequalities is crucial for finding the solution set.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval (a, b) includes all numbers between a and b, but not a and b themselves.

Solving Linear Inequalities

Solving linear inequalities involves finding the values of the variable that satisfy the inequality. This process often includes isolating the variable on one side and may require reversing the inequality sign when multiplying or dividing by a negative number. Mastery of this concept is essential for accurately determining the solution set.