What is the equation equivalent to?
2 answers:
To turn 
into an absolute value inequality we do the following:
First get the average between 1.3 and 2.1. To do this we add 1.3 to 2.1 and then divide by two.
Now that we have the average we can start to create the absolute inequality
Insert our average in the middle
Subtract subtract 1.3 from 1.4 and 1.4 from 2.1 which both will = 0.4. Now setup your absolute value.
First get the average between 1.3 and 2.1. To do this we add 1.3 to 2.1 and then divide by two.
Now that we have the average we can start to create the absolute inequality
Insert our average in the middle
Subtract subtract 1.3 from 1.4 and 1.4 from 2.1 which both will = 0.4. Now setup your absolute value.
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Start with the answer format we want, and work your way toward forming a single fraction like so
Compare that last expression to (2x+1)/(x+2). Notice how the ax and 2x match up, so a = 2 must be the case.
Then we have 2a+b as the remaining portion in the numerator. Plugging in a = 2 leads to 2a+b = 2*2+b = 4+b. Set this equal to the +1 found in (2x+1)/(x+2) to have the terms match.
So, 4+b = 1 leads to b = -3
Therefore, a = 2 and b = -3
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An alternative route:
I added and subtracted 4 in the third step so that I could form 2x+4, which then factors to 2(x+2). That way I could cancel out a pair of (x+2) terms toward the very end.
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Other alternative methods involve synthetic division or polynomial long division. They are slightly separate but related concepts.