If `A + B gt C + D` and `D + A lt B + C`, then

Options

A. `D gt B`
B. `C gt D`
C. `A gt D`
D. `B gt D`

Answer

To solve the problem step by step, we will analyze the given inequalities and manipulate them to find the relationship between the variables.

### Step 1: Write down the given inequalities
We have two inequalities:
1. \( A + B > C + D \) (Inequality 1)
2. \( D + A < B + C \) (Inequality 2)

### Step 2: Rearrange Inequality 1
We can add \( D \) to both sides of Inequality 1:
\[ A + B + D > C + D + D \]
This simplifies to:
\[ A + B + D > C + 2D \quad \text{(Inequality 3)} \]

### Step 3: Rearrange Inequality 2
Now, we can add \( B \) to both sides of Inequality 2:
\[ D + A + B < B + C + B \]
This simplifies to:
\[ A + B + D < 2B + C \quad \text{(Inequality 4)} \]

### Step 4: Compare Inequalities 3 and 4
Now we have:
1. \( A + B + D > C + 2D \) (Inequality 3)
2. \( A + B + D < 2B + C \) (Inequality 4)

From these two inequalities, we can write:
\[ C + 2D < A + B + D < 2B + C \]

### Step 5: Isolate \( D \)
From the inequalities, we can compare the two sides:
\[ 2B + C > C + 2D \]
Subtract \( C \) from both sides:
\[ 2B > 2D \]
Now, divide both sides by 2:
\[ B > D \]

### Conclusion
Thus, we conclude that \( B \) is greater than \( D \).