https://file.aiquickdraw.com/mathbot/file/1729223523653hcyvj2w2.jpg

The expression in the image is:

\[
2 + \frac{3m - n}{5} - \frac{1 - m}{2}
\]

To simplify or manipulate this expression, we can:

1. Find a common denominator between the fractions. The denominators are 5 and 2, so the least common denominator (LCD) is 10.
2. Rewrite each term with the denominator 10:
   \[
   2 + \frac{2(3m - n)}{10} - \frac{5(1 - m)}{10}
   \]
   This becomes:
   \[
   2 + \frac{6m - 2n}{10} - \frac{5 - 5m}{10}
   \]
3. Combine the fractions:
   \[
   2 + \frac{(6m - 2n) - (5 - 5m)}{10}
   \]
   Simplify the numerator:
   \[
   2 + \frac{6m - 2n - 5 + 5m}{10} = 2 + \frac{11m - 2n - 5}{10}
   \]
4. The simplified expression is:
   \[
   2 + \frac{11m - 2n - 5}{10}
   \]
   This can be rewritten as:
   \[
   \frac{20}{10} + \frac{11m - 2n - 5}{10} = \frac{20 + 11m - 2n - 5}{10} = \frac{11m - 2n + 15}{10}
   \]

Thus, the final simplified expression is:
\[
\frac{11m - 2n + 15}{10}
\]

Would you like more details on any step or have any questions?

Here are some related questions to expand on this:

1. How can we solve this expression for \(m\)?
2. How can we solve this expression for \(n\)?
3. What happens if \(m\) and \(n\) are given specific values?
4. How would you handle this expression if additional terms were added?
5. Can this expression be factored further for certain values of \(m\) and \(n\)?

**Tip:** Always look for the least common denominator (LCD) when working with fractions—it helps to combine terms efficiently!

Learn how to simplify the algebraic expression 2 + (3m - n)/5 - (1 - m)/2 using common denominators and fraction operations. This detailed solution includes the least common denominator method and step-by-step combination of terms. Suitable for students in Grades 9-10.

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