1. Consecutive Integers: Numbers are one apart
1st
2nd
3rd
4th
x, x + 1, x + 2, x + 3, …
Consecutive Odd and Even Integers: Numbers are two apart
1st
2nd
3rd
4th
x, x + 2, x + 4, x + 6, …

2. The two integers are 7 and 8.
# 1.
Find two consecutive integers such that the sum of the smaller and 2 times the larger is 23.
x = first integer
x + 1 = second integer
x + 2(x + 1) = 23
The two integers are 7 and 8.
3 x + 2 = 23
3 x = 21
x = 7

3. The two integers are 58 and 60.
# 2.
Find two consecutive even integers such that the sum of the larger and 3 times the smaller is 234.
x = first integer
x + 2 = second integer
x x = 234
The two integers are 58 and 60.
4 x + 2 = 234
4 x = 232
x = 58

4. The two integers are 11and 13.
# 3.
Find two consecutive odd integers such that the sum of the twice larger and 3 times the smaller is 59.
x = first integer
x + 2 = second integer
2(x + 2) + 3 x = 59
The two integers are 11and 13.
5x + 4 = 59
5x = 55
x = 11

5. The two integers are 37 and 38.
# 4.
Find two consecutive integers such that the sum of twice the smaller and the larger is 112.
x = first integer
x + 1 = second integer
(x + 1) + 2 (x) = 112
The two integers are 37 and 38.
3x + 1 = 112
3x = 111
x = 37

6. The three integers are – 11, - 9 and - 7.
# 5.
Find three consecutive odd integers such that the sum of 7 times the smallest and twice the largest is – 91.
x = first integer
x + 2 = second integer
x + 4 = third integer
The three integers are – 11, - 9 and - 7.
7 x + 2 (x + 4) = - 91
7 x + 2 x + 8 = - 91
9 x + 8 = - 91
x = - 11
9 x = - 99

7. # 6. x = first integer x + 1 = second integer x + 2 = third integer
Find three consecutive integers such that the sum of the 3 times smallest and the largest is 54.
x = first integer
x + 1 = second integer
x + 2 = third integer
The three integers are 13, 14, 15
3x + (x + 2) = 54
4x + 2 = 54
4x = 52
x = 13

8. # 7.
# 8.
There are four consecutive integers such that the sum of the first two doubled is the same as the sum of the twice the last one and the second to last one. What is the third integer?
There are four consecutive odd integers such that the sum of the first two is 8 less than the sum of the last two. What is the first integer?
x + (x + 2) = (x + 4) + (x + 6) – 8
2*(6 + 7) = 8 + 2*9
2 x + 2 = 2 x + 2
Third Integer = 8
0 = 0
Since the variables cancel out, and the statement is true, any integers will work.