Solving Linear Inequalities and Graphing Solution Sets

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

-8x ≥ 72

Background

Topic: Linear Inequalities

This question tests your understanding of how to solve linear inequalities, especially when multiplying or dividing both sides by a negative number, and how to represent the solution both graphically and in interval notation.

Key Terms and Formulas

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

• Multiplication Property of Inequality: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

• Interval Notation: A way to describe the set of solutions using parentheses and brackets. For example, (a, b] means all numbers greater than a and up to and including b.

Step-by-Step Guidance

1. Start by identifying the inequality: $-8x \geq 72$.

2. To isolate $x$, divide both sides of the inequality by $-8$. Remember: Dividing by a negative number reverses the direction of the inequality.

3. Write the new inequality after dividing: $x \leq \frac{72}{-8}$.

4. Simplify the right side to find the value that $x$ is compared to.

5. Think about how to graph this solution on a number line: for $x \leq a$, shade to the left of $a$ and use a closed circle at $a$.

Try solving on your own before revealing the answer!

Final Answer: $x \leq -9$; Interval Notation: $(-\infty, -9]$

After dividing and simplifying, you find $x \leq -9$. The interval notation $(-\infty, -9]$ means all real numbers less than or equal to $-9$.