Let's solve the given inequality step by step:

Problem Statement

We need to solve the inequality: $4x + 3 > 8$

Step 1: Solve the inequality

1. Subtract 3 from both sides: $4x > 8 - 3$ $4x > 5$

2. Divide both sides by 4: $x > \frac{5}{4}$

Step 2: Graph the solution on a number line

The solution $x > \frac{5}{4}$ represents all values greater than $\frac{5}{4}$. On the number line, this would be an open circle at $\frac{5}{4}$ with a line extending to the right, indicating all values greater than $\frac{5}{4}$.

Step 3: Write the solution in set-builder notation

The set-builder notation for the solution is: $\{ x \mid x > \frac{5}{4} \}$

Step 4: Write the solution in interval notation

The interval notation for this solution is: $\left( \frac{5}{4}, \infty \right)$

Answer Review

The correct answers are as follows:

• Part b (Set-builder notation): The answer is $\{ x \mid x > \frac{5}{4} \}$, which matches option B.
• Part c (Interval notation): The answer is $\left( \frac{5}{4}, \infty \right)$, which corresponds to option A.

Would you like more details on any of these steps or explanations?

Here are some related questions to consider:

1. How do you determine if a number should have an open or closed circle on a number line?
2. What is the difference between set-builder notation and interval notation?
3. How do you solve inequalities involving variables on both sides?
4. What are the rules for multiplying or dividing inequalities by a negative number?
5. How does graphing an inequality differ from graphing an equation?

Tip: Always remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign!