# Simplify the expression [4-(3+x)/x]/[(x-1)/(x+1)]-[4/(x-x^2)]

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Due to the limitations of the editor box, I have chosen to write at the bottom of the answer my notes explaining, as far as possible, each step. This is in the hope of not disrupting the flow of working that follows. It also allows the reader a chance to deduce how each step leads to the next:

[ 4 - (3+x)/x ] / [ (x-1)/(x+1) ] - [ 4 / (x-x^2) ]

= [ 4 - (3+x)/x ] * (x+1)/(x-1) - [ 4 / (x-x^2) ]

= ( 4x - 3 - x ).(x+1) / [ x.(x-1) ] - 4 / [x (1-x)]

= (3x-3)(x+1) / [ x.(x-1) ] + 4 / [x (x-1)]

= [ 3x^2 - 3x + 3x - 3 + 4 ] / [x.(x-1)]

= **(3x^2 + 1) / [x.(x-1)]** or **(3x^2 + 1) / (x^2 - x)**

Detailed explanation for steps:

Step 1: The idea is a/(b/c) = a * c/b

Step 2: converting 4 into 4x/x and factorizing (x^2 -x ) into x(x-1)

Step 3: simplification of 1st term and converting "-4/[x (1-x)]" into "+4/[x(x-1)]

Step 4 & 5: self explanatory

In order to simplify the given expression, we'll respect the brackets rule.

For the first term of the expression:

[4-(3+x)/x] /[ (x-1)/(x+1)]

The numerator of the first term is 4-(3+x)/x and the denominator is (x-1)/(x+1). To write the numerator in a simplified manner, we have to have the same denominator, which is x. For this reason, we'll amplify 4 with x. The numerator will be:

(4x-3-x)/x=(3x-3)/x=3(x-1)/x

The numerator 3(x-1)/x will be multiplied with the inversed ratio(x+1)/(x-1).

[3(x-1)/x]*[(x+1)/(x-1)],

We'll simplify the common factor (x-1),and we'll get:

3(x+1)/x

This result will be added to the second term -[ 4/(x-2x^2)]

[3(x+1)/x] - [ 4/(x-2x^2)]

[3(x+1)/x]-[ 4/x(1-2x)]

We've noticed that the LCD of the 2 ratios is

x(1-2x), so we'll multiply the first ratio by (1-2x).

The expression will be:

E(x)=[3(x+1-2x^2-2x)-4]/x(1-2x)

We'll combine like terms:

E(x)=(-3x-6x^2-1)/x(1-2x)

We'll re-arrange the terms:

**E(x)=(6x^2+3x+1)/x(1-2x)**

To simplify [4-(3+x)/x]/[(x-1)/(x+1)]-[4/(x-x^2)].

First term numerator : [4-(3+x)/x ) = (4x-3 -x)/x = 3(x-1)/x.

Therefore the first term = {3(x-1)/x}/[(x-1)/(x+1] = 3(x+1)/x .

3 (x+1)/x - 4/(x-x^2) = 3(x+1)/x - 4/x(x-1)

3 (x+1)/x - 4/(x-x^2) = {3(x+1)(x-1) - 4}/x(x-1) = {3x^2 -3 -4}/x(x-1)

3 (x+1)/x - 4/(x-x^2) = (3x^2-7)/[x(x-1)].

Therefore the simlified form of the given expression,

[4-(3+x)/x]/[(x-1)/(x+1)]-[4/(x-x^2)] is (3x^2-7)/[x(x-1)].