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 $ax + b \geq c$.

• Isolate $x$: Use inverse operations (subtract, add, divide, multiply) to get $x$ alone on one side of the inequality.

Step-by-Step Guidance

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

2. Subtract $2$ from both sides to begin isolating $x$:

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

3. Simplify both sides to get $5x$ by itself:

   $5x \geq$ (simplified right side)

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

   $x \geq$ (right side divided by $5$)

Try solving on your own before revealing the answer!

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

Background

Topic: Graphing Solutions to Inequalities on a Number Line

This question tests your ability to interpret and graph the solution set of an inequality on a number line, specifically for $x \leq 3$.

Key Terms and Concepts:

• Closed Circle: Used on a number line to indicate that the endpoint is included ("less than or equal to" or "greater than or equal to").

• 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: $3$ should be marked on the number line.

2. Since the inequality is $\leq$, the circle at $3$ should be closed (filled in).

3. The arrow should extend to the left, indicating all values less than or equal to $3$.

4. Compare each graph option to these criteria to determine which matches $x \leq 3$.

Try solving on your own before revealing the answer!