find three consecutive even intergers for which the sum of the first two integers is greater than 9 more than the third integer, and the sum of the first and third integers is less than 17 more than the second integer
"find three consecutive even intergers"
Let M be the middle integer.
The three numbers are M-2, M, M+2
"the sum of the first two integers is greater than 9 more than the third integer"
(M-2 + M) > M+2 + 9
on solving
M > 13
"sum of the first and third integers is less than 17 more than the second integer"
M-2 + M+2 < M+17
on solving
M < 17
That leaves us with two choices,
M=14, or M=16.
Check:
M=14
12+14=26 > 16+9=25 OK
12+16=28 < 14+17=31 OK
M=16
14+16=30 > 18+9=27 OK
14+18=32 < 16+17=33 OK
So there are two valid solutions,
12,14,16 or 14,16,18
To find three consecutive even integers that satisfy the given conditions, we can follow these steps:
Step 1: Assume the first even integer as 'x'.
Step 2: Since the integers are consecutive, the second even integer would be 'x + 2'.
Step 3: Similarly, the third even integer would be 'x + 4'.
Now, let's set up the equations based on the given conditions:
Condition 1: The sum of the first two integers is greater than 9 more than the third integer.
x + (x + 2) > (x + 4) + 9
Condition 2: The sum of the first and third integers is less than 17 more than the second integer.
x + (x + 4) < (x + 2) + 17
Now, let's solve these equations step by step:
Condition 1:
x + (x + 2) > (x + 4) + 9
2x + 2 > x + 13 --> Simplifying the left side
x > 11 --> Subtracting 'x' from both sides
Condition 2:
x + (x + 4) < (x + 2) + 17
2x + 4 < x + 19 --> Simplifying the left side
x < 15 --> Subtracting 'x' from both sides
From Condition 1, we have x > 11.
From Condition 2, we have x < 15.
Combining these inequalities, we get:
11 < x < 15
Since x represents an even integer, the possible values for x could be 12, 14.
Taking x = 12:
The three consecutive even integers would be 12, 14, 16.
Taking x = 14:
The three consecutive even integers would be 14, 16, 18.
Therefore, the two possible solutions for three consecutive even integers satisfying the given conditions are:
1) 12, 14, 16
2) 14, 16, 18