Oh since its christmas... hehe heres the answer x^2-y^2+3x=0 within the region 2xy + 3y > 0. As you can see, mark initially got this right, however he simplified the inequalities wrong... he gave the final inequality of x>-3/2 Which is not really the simplfied version of 2xy + 3y > 0
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hmm interesting aint it. Anyways i know many of you would think wtf? this guy is nuts why isnt that the simplified version. well good question but it also raises the question of whether you should relearn your inequalities or not. Maybe not, i was jk you dont have to relearn it, this inequality is quite the complicated one to be in the hsc. All you'll need to know is the algebra of harder one dimension inequalities + the graphical drawing of simple two dimension inequalities.
Anyways if you dont believe that the inequality cant be simplified into that just x>-3/2. I shall give you a simple proof~[damn my number 1 button aint working hence no exclamation mark] test the point [-2,-1] into the equations. For those lazy people here it is for both eqN.
For 2xy + 3y > 0
-> 2[-2][-1]+3[-1]>0
= 4-3>0
= 1>0
Is this true? well duh 1 is obviously greater than 0.
But if we test that exact same point into marks given region x>-3/2
-2>-3/2
If thats true in your world mark, im sorry i bothered you, but i wonder if you can lend me the portal to your world, it seems fun there.
Anyways if you still dont believe me, merry christmas. Do not proceed.
OKAY for an alternative answer, we know that in the second quadrant the answers will turn obtuse if added[as said before], in the first quadrant the answer is correct hence let the angles be A and B[in the first quadrant] hence A+B=90. in the second quadrant the angles will be 180-A and 180-B. Hence when they add it will be 180-A+180-B=360-[A+B]
= 360-90
=-pi/2 in the domain of -pi<x<pi
So it isnt in the 2nd quadrant.
In the 4th quadrant it will be -A+-B=-90
=-pi/2
Hence that is not true either.
For the 3rd quadrant it will be A-180 + B-180 = A + B - 360
= 90 - 360
= -270
=pi/2
Hence the hyperbole curve is only correct in the 3rd and 1st quadrant not including the axis.
Remember to get this, we must establish that A+B=pi/2 is true in the first qudrant.
It is recommended that you do the question in the second method as oppose to marks method. Why? easy, because harder inequalities with 2+variables are long and tedious, its takes the same time as doing two of these questions and plus its errorneous. The second method is boring, yes it does look boring and it took me long to type, thats cause my keyboard is broken. Writing this on paper takes 5seconds. Also it is safe, and easily understood by teachers. I mean what if you did wrote 2xy + 3y > 0 down but teacher didnt understand and simplified like mark and gave you the wrong mark, haha there goes another pun. Excuse me.
Also you may use the brute forcing method but you must test the qudrants.
For those who may not know what arctan is, its just inverse tan.
Of course, there are many other ways to do this, but follow the second method and thats all you'll need. If anyone else finds another method please post to show your intelligence.
Merry christmas, im going to melbourne~ yay.
