Question

How is 1/x + 2 - 4/2x +1 equivalent to - 2x + 7 / (x + 2)(2x + 1)

Answer 1

We have,


$\frac{1}{x + 2} - \frac{4}{2x + 1}$

Since the denominator isn't same we cannot directly perform any arithmetic operation on the terms, in order to be able to do that we must make the denominator same,

It's like the way we have,
$\frac{2}{3} + \frac{4}{9}$

You know you cannot just add the terms and write the answer as,
$\frac{8}{12}$
It completely wrong.

each of the terms have different values when when added doesn't gives 8/12.

Let me prove that to you by converting them into decimal values,

so
$\frac{2}{3} = 0.67 \\ \\ and \\ \\ \frac{4}{9} = 0.43$
Adding these two gives, 1.1

and
The result we had earlier equates to,
$\frac{8}{12} = \frac{4}{3} = 1.33$

You see the results are not equal.

So there must be something wrong.

And the wrong this here was not making the denominators equal,
To do that you basically multiply 2/3 with a 3/3 which may look stange at first but it does gives you same denominator as 4/9 in the form of 6/9.

Now you can add the numerators up and leave the denominator as it is which will give 10/9 = 1.1

Back to you question to show that the terms sperated by = are equal we must show one is identical to another.


Let's do this using the Right hand side,

$\frac{( - 2x + 7)}{(x + 2)(2x + 1)} \\ \\ = \frac{(2x + 1) -4 (x + 2)}{(x + 2)(2x + 1)} \\ \\ = \frac{(2x + 1)}{(x + 2)(2x + 1)} - \frac{4(x + 2)}{(x + 2)(2x + 1)} \\ \\ \frac{1}{x + 2} - \frac{4}{2x + 1}$




Now this value is clearly equal to the the one on the left hand side of "=" .