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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1:47 minutes

Problem 34

Textbook Question

Solve each inequality. Give the solution set in interval notation. See Example 4. 2>-6x+3>-3

Verified step by step guidance

Start by breaking the compound inequality into two separate inequalities: $2 > -6x + 3$ and $-6x + 3 > -3$.

Solve the first inequality $2 > -6x + 3$ by subtracting 3 from both sides to isolate the term with $x$: $2 - 3 > -6x$.

Simplify the inequality to get $-1 > -6x$. Then, divide both sides by -6, remembering to reverse the inequality sign: $\frac{-1}{-6} < x$.

Solve the second inequality $-6x + 3 > -3$ by subtracting 3 from both sides: $-6x > -6$.

Divide both sides by -6, again reversing the inequality sign: $x < 1$. Combine the solutions from both inequalities to express the solution set in interval notation.

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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 statements that express the relationship between two expressions that are not necessarily equal. They use symbols such as '>', '<', '≥', and '≤' to indicate whether one side is greater than, less than, or equal to the other. Understanding how to manipulate and solve inequalities is crucial for finding solution sets.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses brackets [ ] to include endpoints and parentheses ( ) to exclude them. For example, the interval (2, 5] includes all numbers greater than 2 and up to and including 5. This notation is essential for expressing the solution set of inequalities clearly.

Solving Compound Inequalities

A compound inequality consists of two or more inequalities that are combined into one statement, often using 'and' or 'or'. To solve a compound inequality, one must isolate the variable while maintaining the relationship between the inequalities. This process often involves breaking the compound inequality into simpler parts and solving each part individually.