If A + D = C + E ,B + 2D = C + 2E ,D - E > E - B And 2D > A + C then                                          C > B > D > A > E

We are given the following system of inequalities and equations:

1. \( A + D = C + E \)
2. \( B + 2D = C + 2E \)
3. \( D - E > E - B \)
4. \( 2D > A + C \)

We are asked to analyze whether the inequality \( C > B > D > A > E \) holds.

Let's proceed step by step:

### Step 1: Analyze the Equations
Starting with the two equations:

- \( A + D = C + E \)
- \( B + 2D = C + 2E \)

From the first equation \( A + D = C + E \), we can rearrange to express \( C \) in terms of the other variables:

\[
C = A + D - E
\]

From the second equation \( B + 2D = C + 2E \), rearrange to express \( B \) in terms of the other variables:

\[
B = C + 2E - 2D
\]

Substituting \( C = A + D - E \) into the equation for \( B \):

\[
B = (A + D - E) + 2E - 2D
\]
\[
B = A - D + E
\]

So now we have two key relationships:

- \( C = A + D - E \)
- \( B = A - D + E \)

### Step 2: Analyze the Inequalities
Now let's analyze the inequalities:

- \( D - E > E - B \)
  Substituting \( B = A - D + E \) into this inequality:
  \[
  D - E > E - (A - D + E)
  \]
  Simplifying:
  \[
  D - E > -A + D
  \]
  Cancel out \( D \) from both sides:
  \[
  -E > -A
  \]
  Which simplifies to:
  \[
  A > E
  \]

- \( 2D > A + C \)
  Substituting \( C = A + D - E \) into this inequality:
  \[
  2D > A + (A + D - E)
  \]
  Simplifying:
  \[
  2D > 2A + D - E
  \]
  Subtract \( D \) from both sides:
  \[
  D > 2A - E
  \]

### Step 3: Conclude the Order of the Variables
Given the relationships derived, we know that:
- \( A > E \)
- \( D > 2A - E \)

Given this information, we can try to compare the variables:

1. Since \( A > E \), \( E \) is the smallest variable.
2. From \( D > 2A - E \), we see that \( D \) is larger than \( 2A - E \).
3. Since \( C = A + D - E \) and \( B = A - D + E \), \( C \) will likely be larger than \( B \) because \( D > 2A - E \).

So the inequality \( C > B > D > A > E \) could hold, given the relationships between the variables. However, for absolute certainty, we'd need specific values to ensure that these comparisons hold true universally.

### Summary
The inequality \( C > B > D > A > E \) is likely to hold under these conditions, based on the derived relationships. The exact order depends on the values of \( A \), \( D \), and \( E \).

Would you like to go into more details or have any further questions? Here are some related questions that may expand your understanding:

1. What happens to the inequality if the relationship \( D - E \) changes?
2. How does the inequality change if \( A \) and \( E \) become closer in value?
3. What if the condition \( 2D > A + C \) does not hold, how does it affect the solution?
4. Can the inequality \( C > B > D > A > E \) be reversed under certain conditions?
5. How would the system change if additional constraints were added to \( B \)?

**Tip:** When solving systems of inequalities, carefully substitute and simplify each inequality to derive relationships between the variables!

If A + D = C + E ,B + 2D = C + 2E ,D - E > E - B And 2D > A + C then C > B > D > A > E

Explore how to solve a system of equations and inequalities involving variables A, B, C, D, and E. The problem utilizes key algebraic concepts such as substitution and properties of inequalities. Suitable for high school students in Grades 10-12.

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