Solving Linear Inequalities and Writing Solution Sets in Interval Notation

Q1. Solve the following inequality and write the solution set in interval notation:

$9(x - 2) < 2(5x - 1)$

Background

Topic: Linear Inequalities

This question tests your ability to solve linear inequalities, which involves using algebraic properties to isolate the variable and then expressing the solution set using interval notation.

Key Terms and Formulas

• Linear Inequality: An inequality that involves a linear expression, such as $ax + b < cx + d$.

• Distributive Property: $a(b + c) = ab + ac$

• Interval Notation: A way to describe the set of solutions using parentheses (for exclusion) and brackets (for inclusion).

Step-by-Step Guidance

1. Apply the distributive property to both sides to eliminate parentheses: $9(x - 2) = 9x - 18$ $2(5x - 1) = 10x - 2$

2. Rewrite the inequality with the distributed terms: $9x - 18 < 10x - 2$

3. Move all terms involving $x$ to one side and constants to the other. Subtract $9x$ from both sides: $9x - 18 - 9x < 10x - 2 - 9x$

4. Combine like terms to simplify: $-18 < x - 2$

5. Isolate $x$ by adding $2$to both sides:$-18 + 2 < x - 2 + 2$

Try solving on your own before revealing the answer!