solving inequalities

Question

the sum of three consecutive even integers is at least 80 and at most 90. what is the largest possible combination of these three integers?

Answer 1

The sum of 3 consecutive even integers is at least 80 and at most 

Call the first number X. The second will be X+2 (the next even number). The third is (X+2)+2=X+4. So the sum is X+X+2+X+4= 3X+6

So we now have 80 ≤ 3x+6 ≤ 90

80 -6≤ 3X+6-6≤ 90-6

74≤ 3X≤ 84

74/3≤ 3X/3≤ 84/3

24.666≤ X≤ 28

Since we know X is an even integer, the only options we have are 26 and 28. 

We already said the X is the smallest number, and the next two are X+2 and X+4. Plugging in the 2 options for x, we have two possible solutions.

1) 26+28+30= 84

2) 28+30+32=90

Answer 2

We have three consecutive even numbers; the smallest one, for convenience, will be called x. The second number is two larger than the smallest, so it is called x + 2. The third (and largest) number is two greater than the second, so it is (x + 2) + 2 or x + 4.

Now, we are asked to give the largest three consecutive even numbers such that their sum is at most 90. Therefore, we are asked mathematically that x + (x + 2) + (x + 4) ≤ 90.

x + (x + 2) + (x + 4) ≤ 90;

3x + 6 ≤ 90;

3x ≤ 84;

x ≤ 28.

The smallest number is 28, so the other two numbers are 30 and 32. This makes the final answer to your solution {28, 30, 32}.

Answer 3

Arithmetic:

(80 / 3 ≈ 26.7  and  90 / 3 = 30) ===> 26 , 28 and 30

Algebra:

If     1^st even number is "x" ,
then 2^nd even number will be "x + 2"
and  3^rd even number will be "x + 4" [(x + 2) + 2]

   80 ≤ x + x + 2 + x + 4 ≤ 90

   80 ≤ 3x + 6 ≤ 90
  - 6          - 6    - 6

   74  ≤  3x  ≤  84 
  ÷ 3    ÷ 3     ÷ 3

   26 ≤ x ≤ 28
Thus, the number are  {26, 28, 30}

Answer 4

First you need to find out what the consecutive integers are for the lowest sum 80.  The equation should be written below to represent three consecutive integers, whose sum = 80.

X + (x+1) + (x+2) = 80

X + x + 1 + x + 2 = 80

3x + 3 = 80

3x = 77

X=25.666

So the next integer is 26. The three consecutive integers are 26+27+28 = 81.

Then you would solve for 3x+3=90, which is the greatest sum.

3x + 3 = 90

3x = 87

X=29

So the integer is 29. The three consecutive integers are 29+30+31=90.

In between these two sets of consecutive integers, there are only a few combinations, you can simply list them and then count how many there are.

1. 26 + 27 + 28 = 81

2. 27 + 28 + 29 = 84

3. 28 + 29 + 30 = 87

4. 29 + 30 + 31 = 90

So there are FOUR possible combinations.