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 interval notation.

Key Terms and Formulas:

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

• Solving Inequalities: Similar to solving equations, but pay attention to the direction of the inequality sign.

Step-by-Step Guidance

1. $\text{Start by isolating the variable } x \text{ in the inequality: } 5x + 2 \geq 12$

2. $\text{Subtract 2 from both sides to get: } 5x \geq 10$

3. $\text{Divide both sides by 5 (since 5 is positive, the inequality direction stays the same): } x \geq 2$

Try solving on your own before revealing the answer!

Final Answer: x \geq 2

This means all values of $x$ that are 2 or greater satisfy the inequality.

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 Formulas:

• Number Line: A visual representation of numbers in order.

• Closed Circle: Used to indicate that the endpoint is included (e.g., $x \leq 3$ means 3 is included).

• Arrow: Shows the direction of the inequality (left for $\leq$, right for $\geq$).

Step-by-Step Guidance

1. $\text{Identify the inequality: } x \leq 3$

2. $\text{On a number line, this means all values to the left of 3, including 3 itself, are part of the solution.}$

3. $\text{Look for a graph with a closed circle at 3 and an arrow pointing left.}$

Try solving on your own before revealing the answer!

Final Answer: Choice b

Choice b shows a closed circle at 3 and an arrow pointing left, which matches $x \leq 3$.