The expression shown in the image is:

$\frac{x}{3} + \frac{x - 2}{6x} - \frac{2 - x}{2x}$

Let’s solve this expression step by step:

Step 1: Find a common denominator

The denominators are $3$, $6x$, and $2x$. To add and subtract the fractions, we need to find a common denominator. The least common denominator (LCD) is $6x$, as it includes all the factors from the individual denominators.

We now express all terms with this common denominator.

1. $\frac{x}{3}$ becomes $\frac{2x}{6x}$.
2. $\frac{x - 2}{6x}$ stays the same.
3. $\frac{2 - x}{2x}$ becomes $\frac{3(2 - x)}{6x}$.

Step 2: Rewrite the expression with the common denominator

Now the expression is: $\frac{2x}{6x} + \frac{x - 2}{6x} - \frac{3(2 - x)}{6x}$

Step 3: Combine the numerators

Since all fractions have the same denominator, we can combine their numerators: $\frac{2x + (x - 2) - 3(2 - x)}{6x}$

Simplify the numerator: $2x + x - 2 - 3(2 - x) = 2x + x - 2 - 6 + 3x = 6x - 8$

Step 4: Write the final simplified expression

The simplified expression is: $\frac{6x - 8}{6x}$

This can be further simplified by factoring out common terms: $\frac{2(3x - 4)}{6x} = \frac{3x - 4}{3x}$

Thus, the simplified result is: $\frac{3x - 4}{3x}$

Would you like more details on any step or further explanation?

Related Questions:

1. How can you find the least common denominator for more complex rational expressions?
2. What happens if one of the variables in the denominator equals zero?
3. Can you factor expressions like $3x - 4$ further in certain cases?
4. How do you solve rational equations involving multiple fractions?
5. What techniques are used to simplify complex fractions with polynomials?

Tip:

Always check for restrictions on the variable, especially if there are variables in the denominator (in this case, $x \neq 0$ to avoid division by zero).