Question 895479
You can solve it this way too.
{{{(2x+1)/(3x-1)=1}}}
{{{2x+1=3x-1}}}
{{{-x=-2}}}
{{{x=-2}}}
So this is one of the critical values.
The other is {{{x=1/3}}} since the denominator goes to zero.
Now split the number line into three regions and test the inequality in those three regions.
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A better way to say it is that by multiplying, you're losing information that's included when the value of the denominator crosses from one sign to the other and it's irretrievable because the assumption you make is that the denominator doesn't go to zero (but in fact it does).
If the denominator never goes to or through zero, you can solve it this way. 
Additionally, allowing for division by zero leads to contradictions. 
Hope that helps.