The equation in the image is:

$\frac{5}{2}x + \frac{1}{2}x = 10 + \frac{1}{2} + \frac{9}{2}x$

Let's solve this step by step.

Step 1: Combine like terms on both sides.

On the left-hand side:

$\frac{5}{2}x + \frac{1}{2}x = \frac{6}{2}x = 3x$

On the right-hand side:

$10 + \frac{1}{2} + \frac{9}{2}x$

First, combine the constants $10 + \frac{1}{2} = \frac{20}{2} + \frac{1}{2} = \frac{21}{2}$, so:

$\frac{21}{2} + \frac{9}{2}x$

Step 2: Set up the simplified equation.

The equation becomes:

$3x = \frac{21}{2} + \frac{9}{2}x$

Step 3: Move the $\frac{9}{2}x$ to the left-hand side by subtracting it.

$3x - \frac{9}{2}x = \frac{21}{2}$

To combine these terms, express $3x$ as $\frac{6}{2}x$:

$\frac{6}{2}x - \frac{9}{2}x = \frac{21}{2}$

$\frac{-3}{2}x = \frac{21}{2}$

Step 4: Solve for $x$.

Multiply both sides by $-2$ to get rid of the fraction:

$-3x = 21$

Finally, divide by $-3$:

$x = \frac{21}{-3} = -7$

Thus, the value of $x$ is $-7$.

Would you like more details or have any questions?

Here are some related questions to explore further:

1. What happens if we modify one of the coefficients in the equation?
2. How can we solve a similar equation with variables on both sides and fractions?
3. What is the general strategy for solving equations involving fractions?
4. How do you check the solution of an equation by substitution?
5. Can you derive equations with more than one variable in a similar fashion?

Tip: Always combine like terms and isolate the variable step by step to simplify equations.