Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -3(x - 6) > 2x - 2

Verified step by step guidance

1. Start by distributing the -3 on the left side of the inequality: write it as \(-3 \times (x - 6)\), which becomes \(-3x + 18\).
2. Rewrite the inequality with the distributed terms: \(-3x + 18 > 2x - 2\).
3. Next, collect all the variable terms on one side and constants on the other. For example, add \$3x\( to both sides and add \)2\( to both sides to isolate \)x$ terms on the right and constants on the left.
4. Simplify both sides after moving terms: you will get an inequality in the form \$18 + 2 > 2x + 3x\(, which simplifies to \)20 > 5x$.
5. Finally, solve for \(x\) by dividing both sides by 5, remembering to keep the inequality direction the same since you are dividing by a positive number. Express the solution set in interval notation based on the inequality you find.

Key Concepts

Solving Linear Inequalities

Linear inequalities involve expressions with variables raised to the first power and inequality signs. To solve them, isolate the variable on one side by performing algebraic operations like addition, subtraction, multiplication, or division, while carefully reversing the inequality sign when multiplying or dividing by a negative number.

Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses, i.e., 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 open intervals (excluding endpoints) and brackets for closed intervals (including endpoints), helping clearly express ranges of values that satisfy the inequality.