To find the value of $x$ when the arithmetic mean of $x + 2$ and $4x + 5$ is $3x + 2$, follow these steps:

Step 1: Write the formula for the arithmetic mean

The arithmetic mean of two numbers $a$ and $b$ is given by: $\text{Arithmetic Mean} = \frac{a + b}{2}$ In this case, the two numbers are $x + 2$ and $4x + 5$.

Step 2: Set up the equation

We know the arithmetic mean of $x + 2$ and $4x + 5$ is equal to $3x + 2$. Therefore, we write: $\frac{(x + 2) + (4x + 5)}{2} = 3x + 2$

Step 3: Simplify the equation

First, simplify the expression inside the parentheses: $\frac{x + 2 + 4x + 5}{2} = 3x + 2$ $\frac{5x + 7}{2} = 3x + 2$

Step 4: Eliminate the fraction

Multiply both sides of the equation by 2 to eliminate the fraction: $5x + 7 = 2(3x + 2)$

Step 5: Expand and simplify

Distribute the 2 on the right side: $5x + 7 = 6x + 4$

Next, move all terms involving $x$ to one side and constants to the other side: $5x - 6x = 4 - 7$ $-x = -3$

Step 6: Solve for $x$

To solve for $x$, divide both sides by $-1$: $x = 3$

Final Answer:

The value of $x$ is $3$.

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Would you like to explore this further or have any questions? Here are some related questions:

1. What are some other methods to solve equations involving arithmetic means?
2. How do you check the correctness of a solution in algebraic equations?
3. What are the applications of arithmetic mean in real-world problems?
4. How does the arithmetic mean compare to other types of means (e.g., geometric mean)?
5. Can an equation involving an arithmetic mean have more than one solution?

Tip: Always verify your final answer by substituting it back into the original equation to ensure it satisfies all parts of the problem.