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 set 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 with the given inequality: $5x + 2 \geq 12$.

2. Subtract $2$from both sides to isolate the term with$x$:

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

3. Simplify both sides:

   $5x \geq 10$

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

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

Try solving on your own before revealing the answer!

Final Answer: $x \geq 2$

Dividing $10$by $5$ gives $2$, so the solution is$x \geq 2$.

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 a number is included in the solution (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 value where the inequality changes: $x = 3$.

2. Since the inequality is $\leq$, use a closed (filled-in) circle at $3$to show that $3$ is included.

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

4. Compare each graph option to see which one matches these criteria.

Try solving on your own before revealing the answer!

Final Answer: Option (b)

Option (b) shows a closed circle at $3$and an arrow to the left, which matches$x \leq 3$.