Find all pairs of consecutive even integers both of which are greater than 7 and their sum is less than 25.

Explanation

To solve the problem of finding all pairs of consecutive even integers that are both greater than 7 and whose sum is less than 25, we can follow these steps:

### Step 1: Define the variables
Let the first even integer be \( x \). Since we are looking for consecutive even integers, the next consecutive even integer will be \( x + 2 \).

### Step 2: Set up the inequalities
According to the problem, both integers must be greater than 7. Therefore, we can write the inequalities:
1. \( x > 7 \) (for the first integer)
2. \( x + 2 > 7 \) (for the second integer, which simplifies to \( x > 5 \), but this is already satisfied by the first inequality)

Additionally, we know that their sum must be less than 25:
\[ x + (x + 2) < 25 \]

### Step 3: Simplify the inequality
Combine the terms in the inequality:
\[ 2x + 2 < 25 \]

### Step 4: Solve for \( x \)
Subtract 2 from both sides:
\[ 2x < 23 \]
Now, divide by 2:
\[ x < 11.5 \]

### Step 5: Combine the inequalities
Now we have two inequalities to consider:
1. \( x > 7 \)
2. \( x < 11.5 \)

### Step 6: Determine the range for \( x \)
Combining these inequalities gives us:
\[ 7 < x < 11.5 \]

### Step 7: Identify the even integers
The even integers that satisfy this inequality are:
- The smallest even integer greater than 7 is 8.
- The next consecutive even integer is 10.

### Step 8: List the pairs of consecutive even integers
The pairs of consecutive even integers greater than 7 and whose sum is less than 25 are:
1. \( (8, 10) \)
2. \( (10, 12) \)

### Conclusion
Thus, the pairs of consecutive even integers that meet the criteria are:
- \( (8, 10) \)
- \( (10, 12) \)