Solve each inequality. Give the solution set using interval notation. -5x - 4≥3(2x-5)

Verified step by step guidance

1. Distribute the 3 on the right side of the inequality: \(3(2x - 5) = 6x - 15\).
2. Rewrite the inequality: \(-5x - 4 \geq 6x - 15\).
3. Add \(5x\) to both sides to start isolating \(x\): \(-4 \geq 11x - 15\).
4. Add 15 to both sides to further isolate \(x\): \(11 \geq 11x\).
5. Divide both sides by 11 to solve for \(x\): \(\frac{11}{11} \geq x\), which simplifies to \(1 \geq x\).

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. Solving inequalities involves finding the values of the variable that make the inequality true.

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 particularly useful for expressing solution sets of inequalities.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to isolate the variable of interest. This includes operations such as adding, subtracting, multiplying, and dividing both sides of an equation or inequality by the same non-zero number. Mastery of these techniques is essential for solving inequalities and understanding their solutions.