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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!

Solve the linear inequality. Then, a. graph the solution on a number line, b. write the solution in set-builder notation, and c. write the solution in interval notation. The inequality is 4x + 3 > 8.

Learn how to solve the linear inequality 4x + 3 > 8 with a detailed step-by-step explanation. This problem covers key algebraic concepts including solving inequalities, graphing on a number line, set-builder notation, and interval notation. Suitable for students in Grades 6-8.

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