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

Verified step by step guidance

1. Start by breaking the compound inequality into two separate inequalities: \$2 > -6x + 3\( and \)-6x + 3 > -3$.
2. Solve the first inequality \$2 > -6x + 3\( by subtracting 3 from both sides to isolate the term with \)x\(: \)2 - 3 > -6x$.
3. Simplify the inequality to get \(-1 > -6x\). Then, divide both sides by -6, remembering to reverse the inequality sign: \(\frac{-1}{-6} < x\).
4. Solve the second inequality \(-6x + 3 > -3\) by subtracting 3 from both sides: \(-6x > -6\).
5. 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.

Key Concepts

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.