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Graphic approach to problems with inequalities

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Re: Graphic approach to problems with inequalities [#permalink] New post 03 Aug 2008, 23:49
Very useful, Walker!
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Re: Graphic approach to problems with inequalities [#permalink] New post 04 Aug 2008, 00:42
Is there any other method to solve these type of questions????
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Re: Graphic approach to problems with inequalities [#permalink] New post 07 Mar 2009, 04:40
walker wrote:
Hi all! My friend, Tarek, PM me and asked me to show how to use the graphic approach to problem with inequalities. I really love such approach because it is not only fast one after training, but also reliable. So, I try to illustrate how to use it.

1) If (x/y)>2, is 3x+2y<18?

(1) x-y is less than 2
(2) y-x is less than 2

1. First of all, we draw x/y>2. x/y=2 - is a boundary. (see figure 1). we should note that if one of the variables is negative and other is positive, x/y will be always negative and less than 2. Therefore, our set of x,y that satisfied x/y>2 lies between line x/y=2 and x-axis.



How do we find which area should be shaded? In first step you defined that set of x and y lies between line x/y=2 and x-axis. Thanks
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Re: Graphic approach to problems with inequalities [#permalink] New post 07 Mar 2009, 08:24
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kbulse wrote:
How do we find which area should be shaded? In first step you defined that set of x and y lies between line x/y=2 and x-axis. Thanks


One of approaches is to check any point.
Let's consider x>0 and try point between x/y=2 and x=0, for example, x=2, y=0.5. For this point x/y=4>2 and the expression is true.
We could also try point above x/y=2 line. For example, x=2, y=2. For this point x/y=1 <2 and the expression is false.
In conclusion, to find what area works, just pick any point in that area (the better to choose a point, for which it is possible to calculate fast).
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Re: Graphic approach to problems with inequalities [#permalink] New post 07 Mar 2009, 09:07
walker wrote:
kbulse wrote:
How do we find which area should be shaded? In first step you defined that set of x and y lies between line x/y=2 and x-axis. Thanks


One of approaches is to check any point.
Let's consider x>0 and try point between x/y=2 and x=0, for example, x=2, y=0.5. For this point x/y=4>2 and the expression is true.
We could also try point above x/y=2 line. For example, x=2, y=2. For this point x/y=1 <2 and the expression is false.
In conclusion, to find what area works, just pick any point in that area (the better to choose a point, for which it is possible to calculate fast).


Hi walker,

how about this way:
x/y>2 => y>x/2 then draw the line y=x/2 and shade everything which is upper side of the line? is it not correct?
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Re: Graphic approach to problems with inequalities [#permalink] New post 07 Mar 2009, 10:05
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kbulse wrote:
how about this way:
x/y>2 => y>x/2 then draw the line y=x/2 and shade everything which is upper side of the line? is it not correct?



1) you should be careful with first operation because it is an inequality and you multiply both sides by y that can have a different sign and change a sign of the inequality.

The correct answer will be:
x/y>2 => x/y * y>2 *y at y>=0 and x/y * y<2 *y at y<0 ==> y<x/2 at y>=0 and y>x/2 at y<0
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Re: Graphic approach to problems with inequalities [#permalink] New post 07 Mar 2009, 10:14
I didn't know that :oops: , i was hesitating whether to ask that question, now I am glad that I asked about that. Thanks
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Re: Graphic approach to problems with inequalities [#permalink] New post 10 Mar 2009, 10:06
Yep, it all makes sense to me now..thanks, really appreciate all the effort.
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Re: Graphic approach to problems with inequalities [#permalink] New post 11 Mar 2009, 10:12
walker wrote:
Thanks, there is a typo here:
Instead of
walker wrote:
... We can put x=4.5 into x-y=2 and find that y=4.25<4.5 (left side).


should be: ... We can put x=4.5 into x-y=2 and find that y=2.5>2.25 (left side, line x-y=2 goes above P).
or even better: ... We can put y=2.25 into x-y=2 and find that x=4.25<4.5 (left side). In other words, line x-y=2 passes y=2.25 (y-coordinate of P) early and goes above P.

I've fixed it in original post.
+1


Hello walker, could you please help me get the final answer? I am clear with the diagram but do not understand how to deduce the answer from it.
intersection of two straight lines is (4.5,2.25).
From first option: x-y<2, I am trying to confirm that whether intersection (4.5,2.25) validates this eqiation.
=> x-y<2
=> y > x-2
=> y > 4.5 - 2
=> y > 2.5
But at x=4.5, actual value of y is 2.25, which is, indeed, not greater than 2.5!! What should I look for from this statement?

I guess I am missing something..... :?

....thank you.
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Re: Graphic approach to problems with inequalities [#permalink] New post 11 Mar 2009, 10:34
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priyankur_saha@ml.com wrote:
...
But at x=4.5, actual value of y is 2.25, which is, indeed, not greater than 2.5!! What should I look for from this statement?

I guess I am missing something..... :? .


Figure 3. We have "true" and "false" regions. What happens in remaining area we don't care as the area doesn't satisfies problem's conditions.
Figure 4. Part of "true/false" regions is gray because it doesn't satisfy x-y<2 condition. Only in "color" part x-y<2 is true.

But in Figure4 we could have doubt about point P: where line x-y<2 passes P, left-above or bottom-right. In first case we will have only "true" region and in second case - "true" and "false" regions. As you correctly pointed out at x=4.5 y=2.5>2.25. So, point P is not included in "color" region (P does not satisfy x-y<2 condition) and x-y<2 passes left-above. So that, we can conclude we have only "true" region.
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Re: Graphic approach to problems with inequalities [#permalink] New post 18 Mar 2009, 19:05
Ok this might be a dumb question, but I see both graphs 4 and 5 lines passing from the true region.
How can you deduce which one is correct?

Also for graph 4 "walker" said, "4. Let's consider fist condition: x-y<2. x-y=2 is a boundary. (see figure 4). As we can see all y,x that satisfies the fist condition lie in "green-TRUE" region."

But I see some points that lie outside, how can "all" points satisfy this?

I m in waiting list, gotta take the test again to get in. Please please can somebody help me :shock: ?
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Re: Graphic approach to problems with inequalities [#permalink] New post 25 Mar 2009, 10:32
Thanks for the clarification walker...... +1 for you..... :-D .... marvellous approach.
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Re: Graphic approach to problems with inequalities [#permalink] New post 04 May 2009, 23:06
Thank you very much. This really helps. :lol:
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Re: Graphic approach to problems with inequalities [#permalink] New post 13 May 2009, 22:02
Hi Walker,

This approach works for linear equation or can we use it for equations like X^2 or X^3 also ??
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Re: Graphic approach to problems with inequalities [#permalink] New post 18 May 2009, 10:24
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mdfrahim wrote:
Hi Walker,

This approach works for linear equation or can we use it for equations like X^2 or X^3 also ??


This approach works for any equations that could be drawn under GMAT time conditions. Y = X^3 + 1 is easy to draw but Y = 0.34X^3 - 342X^2 + 6X - 3 is not.
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Re: Graphic approach to problems with inequalities [#permalink] New post 01 Aug 2009, 23:32
Inequalities have been bane of my life :roll:
But this approach is pure genius:!: and it finally gives me hope :) Thanks a lot.

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Re: Graphic approach to problems with inequalities [#permalink] New post 02 Aug 2009, 03:53
thanks... but i think it is difficult to use ... ven ur pressed for time
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Re: Graphic approach to problems with inequalities [#permalink] New post 02 Aug 2009, 09:32
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For me this approach has one great advantage over other ones: When picture is drawn, the answer is almost obvious. Moreover, drawing is pretty straightforward process: line by line, expression by expression.
Nevertheless, the more approaches you know, the more confident you are.
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Re: Graphic approach to problems with inequalities [#permalink] New post 10 Aug 2009, 16:43
Hi Walker - for those of us who are artistically challenged, do you happen to know the way to solve this algebraically? :)
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Re: Graphic approach to problems with inequalities [#permalink] New post 10 Aug 2009, 22:03
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understudy wrote:
Hi Walker - for those of us who are artistically challenged, do you happen to know the way to solve this algebraically? :)


I will try to solve it using "construct example" approach:

x/y >2, 3x+2y <18?

a) Could we construct an example when 3x+2y<18 ? We need small x and y for witch x/y >2.
Let's say we have x = 1 (I mean 1.000001) and y=0.5

1/0.5 > 2 - Ok
3*1+2*0.5 <18 Ok

first statement: 1-0.5<2 Ok
second statement: -0.5+1<2 Ok

So, x=1.000001 and y=0.5 is an example that satisfies both statements and answer is YES (True) that 3x + 2y < 18

b) Could we construct an example when 3x+2y>=18 ? The condition x/y > 2 says that x and y can be both positive or negative. At negative x,y 3x+2y will be always negative. So, we need look for our example among x,y positive.

first statement: x-y<2. Under this statement we can say that positive x and y must be close to each other. So, let's consider the maximum difference: x-y = 2 ---> y+2/y >2 --> y=2 and x=4 (as earlier I mean x=3.9999999 y=1.99999999).
At such x and y we have 3*4+ 2*2 = 16 < 18. So, we can't construct an example for witch 3x+2y >=18 and first statement is sufficient.

second statement: y-x < 2. Under this statement we can choose x=100000 and y =1 and 3x+2y >=18. So, we can construct an example for witch 3x+2y >=18 and second statement is insufficient.

My comments: This approach depends on concrete problem and your luck to see an examples. At the same time I love graph approach as it is always straightforward.
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Re: Graphic approach to problems with inequalities   [#permalink] 10 Aug 2009, 22:03
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