Question
Sun April 29, 2012 By: Tejaskumar Hiteshkumar Gajjar

Solve 4/(x-3)

Expert Reply
Mon April 30, 2012
4/(x-3)<1
 
To solve this inequality, we will be required to multiply x-3 on both sides, but it depends on whether x-3 is positive or negative. Now, x-3 can be either positive or negative depending on the values of x. Hence we have to assume 2 cases,
Case 1- x-3 is positive, means x-3>0 or x>3.
In this case, we can multiply direct by x-3 on both sides since it is positive
Rearranging we can write the above as
4<x-3
or, x>7
this is according to our assumption that x>3, so this value holds good in this range.
So first half of the solution becomes x > 7 means x = (7,infinity)
 
Now case 2- x-3 is negative, means x-3<0 or x<3.
In this case, we cannot multiply direct by x-3 on both sides since it is negative.
When we multiply by x-3 on both sides, there will be change of sign, since we are multiplying by a negative number.
Rearranging we can write the above as
4>x-3
or, x<7
Now, according to our assumption that x<3, so this value holds good in this range only for x<3.
So second half of the solution becomes x <3 means x = (-infinity, 3)
So the total solution becomes  (-infinity,3)U(7,infinity). 
 
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