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, such as $ax + b \geq c$.

• Solving Inequalities: Similar to 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 to get:

   $5x \geq 10$

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.

Try solving on your own before revealing the answer!

Final Answer: $x \geq 2$

After isolating $x$, you find that $x$ must be greater than or equal to $2$ for the original inequality to be true.

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 $x \leq a$, shade or draw the arrow to the left of $a$.

Step-by-Step Guidance

1. Identify the endpoint on the number line, which is $3$ in this case.

2. Since the inequality is $x \leq 3$, use a closed circle at $3$ to show that $3$ is included in the solution.

3. Shade or draw an arrow to the left of $3$ to represent all values less than or equal to $3$.

4. Compare each answer choice to see which one matches these criteria.

Try solving on your own before revealing the answer!

Final Answer: Choice b

Choice b shows a closed circle at $3$ and the arrow extending to the left, which correctly represents $x \leq 3$.