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 the expression as \(-3 \times (x - 6)\) and apply the distributive property to get \(-3x + 18\).
2. Rewrite the inequality with the distributed terms: \(-3x + 18 > 2x - 2\).
3. Next, collect all the variable terms on one side and the constant terms on the other side. For example, add \$3x\( to both sides and add \)2\( to both sides to isolate terms: \)18 + 2 > 2x + 3x$.
4. Simplify both sides: combine like terms to get \$20 > 5x$.
5. Finally, solve for \(x\) by dividing both sides of the inequality 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 obtained.

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, similar to solving equations, but remember to reverse 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 correctly is essential to simplify expressions before solving inequalities.

Interval Notation

Interval notation is a way to represent solution sets of inequalities using intervals. It uses parentheses for values not included and brackets for values included, clearly showing the range of possible solutions on the number line.