Solve each inequality. Give the solution set in interval notation. 6x-(2x+3)≥4x-5

Verified step by step guidance

1. Start by simplifying both sides of the inequality. Distribute the negative sign on the left side: write \(6x - (2x + 3)\) as \(6x - 2x - 3\).
2. Combine like terms on the left side: \(6x - 2x\) simplifies to \$4x$, so the inequality becomes \(4x - 3 \geq 4x - 5\).
3. Next, isolate the variable terms on one side by subtracting \$4x$ from both sides: \(4x - 3 - 4x \geq 4x - 5 - 4x\).
4. Simplify both sides after subtraction: the left side becomes \(-3\) and the right side becomes \(-5\), so the inequality is \(-3 \geq -5\).
5. Analyze the inequality \(-3 \geq -5\). Since this is a true statement and does not involve \(x\), conclude what this means for the solution set in interval notation.

Key Concepts

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side to find the range of values that satisfy the inequality. Similar to equations, operations like addition, subtraction, multiplication, and division are used, but special care is needed when multiplying or dividing by negative numbers, as this reverses the inequality sign.

Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses, such as a(b + c) = ab + ac. Applying this property simplifies expressions and is essential for removing parentheses before solving inequalities or equations.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using intervals. It uses parentheses for values not included (open intervals) and brackets for values included (closed intervals), clearly showing the range of possible solutions.