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Three numbers are plotted on a number line, as shown.

​ 
This item has three parts.
​
Part A
Which expression is equivalent to m−nm−n?​     

A

−m+n-m+n−m+n 

B

−m+(−n)-m+(-n)−m+(−n) 

C

m+nm+nm+n 

D

m+(−n)m+(-n)m+(−n) 

(b)

    

Part B
For each expression, select a value from the drop-down to identify whether the value of the expression is to the left or to the right of mm on the number line.
​​m+nm+n     

to the right of m

    ​​m−nm−n ​     

to the left of m

    ​m+pm+p ​     

to the right of m

    ​m−pm−p ​     

to the left of m

    
​Part C
Create an expression that represents the distance on the number line between mm and pp.
  

    

1 answer

Let's break this down into the three parts as requested.

Part A We need to simplify and find an expression equivalent to \( m - n - m - n \).

  1. Start with the original expression: \[ m - n - m - n \]

  2. Combine like terms: \[ (m - m) + (-n - n) = 0 - 2n = -2n \]

Now let's analyze the options to find which one simplifies to \(-2n\).

Option A: \(-m + n - m + n - m + n\) becomes: \[ (-m - m - m) + (n + n + n) = -3m + 3n \]

Option B: \(-m + (-n) - m + (-n) - m + (-n)\) becomes: \[ (-m - m - m) + (-n - n - n) = -3m - 3n \]

Option C: \(m + n + m + n\) becomes: \[ 2m + 2n \]

Option D: \(m + (-n) + m + (-n)\) becomes: \[ (m + m) + (-n + -n) = 2m - 2n \]

None of the given options equate to \(-2n\), which seems to suggest we may have misunderstood the expression requested.

However, if we interpret \( m - n - m - n \) properly, we note that we want a reduction which is identical to compact forms of simplified expressions.

Based on breakdown and checking:

  • The equivalent expression doesn’t seem present.

But, based on interpreting your task, if trying \( -n -n \), we can conclude properties of additions is similar.

So, I'll assume next request is about location on the number line; continuing onward.

Part B

  1. For \( m + n \):
  • This is simply more than \( m \); to the right of \( m \).
  1. For \( m - n - m - n \) :
  • This simplifies to a negative or smaller form, thus would be left.
  1. For \( m + p + p \):
  • Because it involves positive addition; again, points to more than \( m \), therefore to the right.
  1. Finally, for \(m - p - m - p\):
  • This resembles reductions and would land you back left.

Thus the completion should clarify where they are positioned in your layout.

Part C For creating an expression that represents the distance between \( m \) and \( p \) on a number line, we can use the absolute value of their difference: \[ |m - p| \] This expression represents the distance between the two points on the number line where \( m \) is positioned, without any loss of generality.

In summary, your final parts reference:

  1. The expression which simplifies isn't clear from options, affirm it's \( m - n - m - n \).
  2. The placements lead to clarifications determining left versus right on additive components, leading to:
    • \( m+n \) right, \( m-n-m-n \) left, and so forth.
  3. Finally concluding \( |m - p| \) summarizes your request well.

If you'd like further clarification or variations, please advise.