To find the value of $x$ given that the arithmetic mean of 3 and $3x + 5$ is 8, we can follow these steps:

1. Understanding the Arithmetic Mean: The arithmetic mean of two numbers $a$ and $b$ is given by: $\text{Mean} = \frac{a + b}{2}$

2. Setting Up the Equation: Given the two numbers are 3 and $3x + 5$, and their arithmetic mean is 8, we can set up the equation: $\frac{3 + (3x + 5)}{2} = 8$

3. Simplifying the Equation: Combine the terms in the numerator: $\frac{3 + 3x + 5}{2} = 8$ $\frac{3x + 8}{2} = 8$

4. Solving for $x$: Multiply both sides of the equation by 2 to eliminate the denominator: $3x + 8 = 16$ Subtract 8 from both sides: $3x = 8$ Divide both sides by 3: $x = \frac{8}{3}$

Therefore, the value of $x$ is $\frac{8}{3}$.

Would you like a detailed explanation or have any questions? Here are some related problems you might find interesting:

1. Find the arithmetic mean of 4 and $2x + 7$ if the mean is 9.
2. Determine $x$ if the arithmetic mean of 5 and $4x - 3$ is 11.
3. Solve for $x$ if the arithmetic mean of $x$ and $2x + 10$ is 15.
4. Find $y$ if the arithmetic mean of 7 and $5y - 4$ is 12.
5. Determine the value of $x$ if the arithmetic mean of $6$ and $2x + 9$ is 10.
6. Solve for $x$ if the arithmetic mean of $8$ and $3x + 6$ is 14.
7. Find $x$ if the arithmetic mean of $10$ and $4x + 2$ is 13.
8. Determine $x$ if the arithmetic mean of $2$ and $3x + 1$ is 7.

Tip: When solving equations involving arithmetic means, remember to clear the fraction by multiplying both sides of the equation by the denominator to simplify the solving process.