Solve the following inequality algebraically `|(x+3)/(x-1)| > 1`



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Posted on (Answer #1)


This will give two initial solutions,

`(x+3)/(x-1) gt 1` or `(x+3)/(x-1) lt-1`

1) `(x+3)/(x-1) gt 1`

`(x+3)/(x-1)-1 gt 0`

`(x+3-(x-1))/(x-1) gt0 `

This gives,


`1/(x-1) gt0`

Multiplying both denominator and numerator by (x-1),

`(x-1)/(x-1)^2 gt0`

This would make the denominator alway positive, and the requirement for this inequality would be,


`x gt1`


2)`(x+3)/(x-1) lt-1`

`(x+3)/(x-1) +1 lt 0`

This gives,


`(2x+2)/(x-1) lt 0`

`(x+1)/(x-1) lt 0`

Multiplying both denominator and numerator by (x-1),

`((x+1)(x-1))/(x-1)^2 lt0`


This gives,

(x+1)(x-1) < 0

The solution for this is -1<x<1


By combining above two solutions we can get the final solution.

Therefore the required solution for `|(x+3)/(x-1)|gt1` is `xgt -1` .


The answer is x > -1


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