Solve each inequality. Give the solution set in interval notation. See Examples 1 and 2. 2-4x+5(x-1)<-6(x-2)

Verified step by step guidance

1. Distribute the terms on both sides of the inequality: \(2 - 4x + 5(x - 1) < -6(x - 2)\).
2. Simplify each side by distributing: \(2 - 4x + 5x - 5 < -6x + 12\).
3. Combine like terms on the left side: \(2 + x - 5 < -6x + 12\).
4. Combine constants on the left side: \(x - 3 < -6x + 12\).
5. Add \(6x\) to both sides to isolate \(x\): \(x + 6x - 3 < 12\).

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 less than, greater 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 parentheses and brackets to indicate whether the endpoints are included (closed interval) or excluded (open interval). For example, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side of the inequality sign, similar to solving linear equations. However, one must be cautious when multiplying or dividing by a negative number, as it reverses the inequality sign. This process often leads to determining the range of values that satisfy the inequality.