Okay, let's see. The problem says, "Half the value of 3x minus 9 is x plus 37. What is the value of x?" Hmm, I need to translate this into an equation and solve for x. Let me start by breaking it down.

First, "half the value of 3x - 9" means I have to take the expression 3x - 9 and multiply it by 1/2. So that part would be (1/2)(3x - 9). Then, this is equal to "x + 37". So putting it all together, the equation should be:

(1/2)(3x - 9) = x + 37

Alright, now I need to solve for x. Let me write that equation again to make sure I have it right. Yeah, (3x - 9)/2 = x + 37. To solve for x, I can start by eliminating the denominator. Maybe multiply both sides by 2 to get rid of the fraction. Let's try that.

Multiplying both sides by 2 gives:

2*(3x - 9)/2 = 2*(x + 37)

Simplifying the left side, the 2 in the numerator and denominator cancels out, leaving 3x - 9. On the right side, it's 2x + 74. So now the equation is:

3x - 9 = 2x + 74

Now, I need to get all the x terms on one side and constants on the other. Let me subtract 2x from both sides to get:

3x - 2x - 9 = 74

Which simplifies to:

x - 9 = 74

Then, add 9 to both sides:

x = 74 + 9

x = 83

Wait, let me check that again. Starting from 3x - 9 = 2x + 74. Subtract 2x: x - 9 = 74. Then add 9: x = 83. Hmm, seems straightforward. Let me verify by plugging x = 83 back into the original equation to make sure it works.

Original equation: (1/2)(3x - 9) = x + 37

Left side: (1/2)(3*83 - 9) = (1/2)(249 - 9) = (1/2)(240) = 120

Right side: 83 + 37 = 120

Yes, both sides equal 120. So x = 83 is correct. I think that's the answer. Let me just recap the steps to ensure I didn't make any mistakes. Starting with the equation, multiplied both sides by 2, simplified, solved for x, and checked the solution. All steps seem valid. Therefore, x must be 83.
Therefore, the value of $x$ is $\boxed{83}$.
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