https://file.aiquickdraw.com/mathbot/file/1727357889548i74cjwh7.jpg

In the provided image, we are given:

- The length of \(MO = 12x\),
- The length of \(MN = 7x + 1.5\),
- The length of \(NO = 2x\).

To find \(MN\), we use the relationship that \(MO = MN + NO\).

So, the equation becomes:
\[
12x = (7x + 1.5) + 2x
\]

Simplifying the right-hand side:
\[
12x = 9x + 1.5
\]

Now, subtract \(9x\) from both sides:
\[
12x - 9x = 1.5
\]
\[
3x = 1.5
\]

Solving for \(x\):
\[
x = \frac{1.5}{3} = 0.5
\]

Now that we know \(x = 0.5\), we can substitute this value into the expression for \(MN = 7x + 1.5\):
\[
MN = 7(0.5) + 1.5 = 3.5 + 1.5 = 5
\]

Thus, \(MN = 5\).

Would you like further explanation or have any additional questions?

Here are 5 related questions to explore further:
1. How would the equation change if \(MO\) were given as \(15x\)?
2. What would \(NO\) be if \(MO = 14x\) and \(MN = 8x + 2\)?
3. Can you find the total length of \(MO\) if \(x = 2\)?
4. How does changing the constant in \(MN = 7x + 1.5\) affect the solution?
5. How would you set up the problem if \(MO = MN + 3NO\)?

**Tip:** When solving for a variable in equations, always substitute the variable back into other expressions to check your solution.

Given that MO=12x and the provided picture, find MN.

This problem involves finding the value of MN based on the equation MO = 12x and the provided expressions for MN and NO. The problem includes solving a linear equation using substitution. Suitable for middle school students in Grades 6-8.

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