Solving Linear Inequalities and Graphing Solutions on a Number Line

Q9. Solve: $5x + 2 \geq 12$

Background

Topic: Linear Inequalities

This question tests your ability to solve a linear inequality for $x$ and interpret the solution in inequality notation.

Key Terms and Formulas

• Linear Inequality: An inequality that involves a linear expression (e.g., $ax + b \geq c$).

• Solving Inequalities: Use similar steps as solving equations, but remember to reverse the inequality sign if you multiply or divide by a negative number.

Step-by-Step Guidance

1. Start by isolating the term with $x$ on one side. Subtract $2$ from both sides of the inequality:

   $5x + 2 - 2 \geq 12 - 2$

2. Simplify both sides:

   $5x \geq 10$

3. Divide both sides by $5$ to solve for $x$:

   $\frac{5x}{5} \geq \frac{10}{5}$

4. Simplify the result to get $x$ by itself. What does the inequality look like now?

Try solving on your own before revealing the answer!

Q10. Which answer choice below is the graph of $x \leq 3$?

Background

Topic: Graphing Inequalities on a Number Line

This question tests your ability to interpret and graph the solution set of an inequality on a number line.

Key Terms and Concepts

• Closed Circle: Used on a number line to indicate that the endpoint is included (for $\leq$ or $\geq$).

• Arrow Direction: For $x \leq 3$, shade or draw the arrow to the left of $3$.

Step-by-Step Guidance

1. Identify the correct endpoint on the number line (at $x = 3$).

2. Determine if the circle at $x = 3$ should be open or closed. For $x \leq 3$, should it be filled in?

3. Check which direction the shading or arrow should go. For $x \leq 3$, should it point left or right?

4. Compare each answer choice to these criteria to find the correct graph.

Try solving on your own before revealing the answer!