April 5th, 2014, 09:17 AM  #1 
Senior Member Joined: Apr 2013 Posts: 425 Thanks: 24  An inequality
To solve the inequality $\displaystyle 3ix^2+4x5i\leq 0$ where $\displaystyle i^2=1$.

April 5th, 2014, 09:52 AM  #2 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
$\displaystyle x\,=\,\sqrt{\frac53}$.

April 5th, 2014, 07:35 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra 
I couldn't disagree more. To me the inequality is meaningless because the complex plane has no ordering. Should there be some modulus symbols somewhere? Is $x$ supposed to be real or complex? 
April 5th, 2014, 08:03 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,974 Thanks: 1156 Math Focus: Elementary mathematics and beyond 
Try subbing in Olinguito's answer.

April 5th, 2014, 09:26 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra 
I know where that came from, and yes, by eliminating the imaginary part of the equation it makes some sort of sense of the inequality. But I would suggest that it's a poorly set question if that is the answer that is being sought.

April 5th, 2014, 10:26 PM  #6 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
Perhaps it was a trick question. 
April 5th, 2014, 11:15 PM  #7 
Senior Member Joined: Apr 2013 Posts: 425 Thanks: 24  
April 6th, 2014, 12:57 AM  #8 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
If you allow $x$ to be complexvalued, you could also have $$x\,=\,\frac{\pm\sqrt{11}+2i}3$$ By the way, was it meant to be a trick question – i.e. was it meant to provoke the reaction “this inquality is meaningless”? If it was, it was perfect! Had you posted it a few days earlier it would have even made an excellent April fool’s joke. PS: I’d like to point out that just because an inequality contains complex numbers does not mean that it must be meaningless. For example, the interior of unit circle can be described by $\displaystyle z\overline z<1$. If this inequality makes sense, despite the presence of complex variables, I don’t see why we should not try to make sense of the OP’s inequality. Last edited by Olinguito; April 6th, 2014 at 01:29 AM. 
April 6th, 2014, 01:47 AM  #9 
Senior Member Joined: Mar 2014 Posts: 112 Thanks: 8  Olinguito is correct because 3ix²  5i = 0 and 4x ≤ 0. The key to solving this problem correctly is to see how to get rid of the imaginary numbers to solve such problem. There could be more solutions but I won't go through all of them. 
April 6th, 2014, 07:09 AM  #10 
Senior Member Joined: Apr 2013 Posts: 425 Thanks: 24  Solving of the inequality $\displaystyle 3ix^2+4x5i\leq 0$ where $\displaystyle i^2=1$ is simple if it is considered an equivalent equation $\displaystyle 3ix^2+4x5i=a$ where $\displaystyle i^2=1$ and $\displaystyle a\in R,a\leq 0$.


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