Solving Linear Inequalities and Expressing Solutions in Interval Notation

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

$\frac{5}{6}x \geq -7$

Background

Topic: Linear Inequalities

This question tests your understanding of how to solve linear inequalities, graph the solution on a number line, and express the solution in interval notation. It also checks your knowledge of how multiplying or dividing both sides of an inequality by a positive number affects the inequality sign.

Key Terms and Formulas

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

• Multiplication Property of Inequality: If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality does not change.

• Interval Notation: A way to describe the set of solutions using brackets [ ] for included endpoints and parentheses ( ) for excluded endpoints.

Step-by-Step Guidance

1. Start by isolating $x$ in the inequality $\frac{5}{6}x \geq -7$. To do this, multiply both sides by the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$.

2. Apply the multiplication property of inequality: $\frac{6}{5} \cdot \frac{5}{6}x \geq \frac{6}{5} \cdot (-7)$.

3. Simplify the left side so that $x$ is by itself, and simplify the right side by multiplying $-7$ by $\frac{6}{5}$.

4. Since you are multiplying both sides by a positive number, the inequality sign remains the same.

5. Express the solution as $x \geq$ (your simplified value from the previous step).

Try solving on your own before revealing the answer!

Final Answer: $\left[ -\frac{42}{5}, \infty \right)$

The solution set includes all $x$ values greater than or equal to $-\frac{42}{5}$, which is shown on the number line with a bracket at $-\frac{42}{5}$ and shading to the right.