What is the equation to represent the relationship among the integers for: 

The greater of two consecutive even integers is six less than twice the smaller.

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The idea is to get all the words out and mathematical symbols in

Let S = the smaller (of two consecutivs integers)

Since the greater is one more than the smaller

we let S + 1 = the greater.

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The greater of two consecutive even integers is six less than twice the smaller.

So replace the red words by S + 1 and the blue words by S.

S + 1 is six less than twice S. 

Now replace " twice S " by " 2S ", and we have

S + 1 is six less than 2S

Replace the word " is " by an equal sign " = ".

S+1 = six less than 2S

The only thing left is to decide how to get rid of the words
"six less than".

Let's think about what, say, "six less than 10" means. It means to start with
10 and then subtract 6, or 10 - 6.  This, of course, would be 4.

Let's also think about what, say,  "six less than 73" means. It means to start
with 73 and the subtract 6, or 73 - 6. This, or course would be 67.

Now let's think about what "six less than 2S" means. It means to start with
2S and subtract 6, or 2S - 6.  We don't know what this would be, so all we can
do is just leave it as 2S - 6.  

So the final equation is

S + 1 = 2S - 6

Edwin