Find the value of x when the arithmetic mean of x+2 and 4x+5 is 3x+2

To find the value of \( x \), we'll first recall that the arithmetic mean of two numbers is given by the formula:

\[
\text{Arithmetic Mean} = \frac{\text{sum of the numbers}}{2}
\]

Here, the arithmetic mean of \( x + 2 \) and \( 4x + 5 \) is given as \( 3x + 2 \).

So, we set up the equation for the arithmetic mean:

\[
\frac{(x + 2) + (4x + 5)}{2} = 3x + 2
\]

Now, let's simplify and solve for \( x \).

### Step 1: Simplify the numerator
\[
(x + 2) + (4x + 5) = x + 4x + 2 + 5 = 5x + 7
\]

### Step 2: Write the equation
\[
\frac{5x + 7}{2} = 3x + 2
\]

### Step 3: Eliminate the fraction by multiplying both sides by 2
\[
5x + 7 = 2(3x + 2)
\]
\[
5x + 7 = 6x + 4
\]

### Step 4: Rearrange the equation to isolate \( x \)
Subtract \( 6x \) from both sides:
\[
5x - 6x + 7 = 4
\]
\[
-x + 7 = 4
\]

### Step 5: Solve for \( x \)
Subtract 7 from both sides:
\[
-x = 4 - 7
\]
\[
-x = -3
\]

Multiply both sides by \( -1 \):
\[
x = 3
\]

Thus, the value of \( x \) is \( \boxed{3} \).

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. What is the arithmetic mean of \( x + 3 \) and \( 5x + 7 \) in terms of \( x \)?
2. How do you find the geometric mean between two numbers?
3. Can you solve a similar problem with \( x + 4 \) and \( 3x + 8 \)?
4. How would you solve for \( x \) if the harmonic mean of two numbers is given?
5. What happens if the arithmetic mean equals one of the two numbers?

**Tip**: Always simplify the equation first before attempting to solve for the unknown variable.

Learn how to find the value of x when the arithmetic mean of x+2 and 4x+5 equals 3x+2. This problem focuses on algebraic concepts, specifically arithmetic mean and linear equations, suitable for students in Grades 8-10.

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