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

Verified step by step guidance

1. Start by writing down the inequality: \(-5x - 4 \geq 3(2x - 5)\).
2. Distribute the 3 on the right side: \(-5x - 4 \geq 6x - 15\).
3. Get all terms involving \(x\) on one side and constants on the other. Add \$5x$ to both sides and add \(15\) to both sides: \(-4 + 15 \geq 6x + 5x\).
4. Simplify both sides: \(11 \geq 11x\).
5. Divide both sides by 11 (note that 11 is positive, so the inequality direction stays the same): \(\frac{11}{11} \geq x\), which simplifies to \(1 \geq x\). This means \(x \leq 1\).

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. This is essential for simplifying expressions on either side of the inequality before solving for the variable.

Interval Notation

Interval notation is a concise way to represent the solution set 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 solutions on the number line.