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# The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| =

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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
I am sorry but I don't understand how do we arrive at the shaded region whose area is required. So, can anyone out there help me out? Thanks.

Regards,
Pallavi
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
pallaviisinha wrote:
I am sorry but I don't understand how do we arrive at the shaded region whose area is required. So, can anyone out there help me out? Thanks.

Regards,
Pallavi

We have to make a graph with 6 equations:
x=1
x=-1
y=1
y=-1
x+y=1
x+y=-1

The shaded region is one with the overlap..two squares of side 1 & two right triangles with area 1/2 each..which gives the answer as 3.

Unfortunately in the photo accompanying the explanation, in place of the line x=-1, the line x=-2 is drawn. Perhaps that caused the confusion.

Hope it helped!
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
ytsejam wrote:
pallaviisinha wrote:
I am sorry but I don't understand how do we arrive at the shaded region whose area is required. So, can anyone out there help me out? Thanks.

Regards,
Pallavi

We have to make a graph with 6 equations:
x=1
x=-1
y=1
y=-1
x+y=1
x+y=-1

The shaded region is one with the overlap..two squares of side 1 & two right triangles with area 1/2 each..which gives the answer as 3.

Unfortunately in the photo accompanying the explanation, in place of the line x=-1, the line x=-2 is drawn. Perhaps that caused the confusion.

Hope it helped![/quoteTh

Indeed, thanks.
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
manpreetsingh86 wrote:
sunny3011 wrote:
The area bounded by the three curves |x + y | = 1, | x | = 1, and | y | = 1, is equal to
a. 1
b. 2
c. 3
d. 4
e. 8

I have tried to plot graph of lines and I got area from |x|=1 and |y|=1. but what about line |x+y|=1???
Confused and need to clarify the concepts?

Source: GMAT Tutor Question bank

|x+y|=1 can be written as x+y=1 or x+y=-1. These are two straight lines, which are mirror image of each as shown in the following graph.

|x|=1 and |y|=1 are also drawn on the graph with red and blue lines respectively. The required area is shown in the grey blocks. which is equal to 3(2+1/2+1/2). (2 block of area 1 and two triangles of area =1/2)

The solution is correct.
However,I think the line x=-1 is not represented properly here...
It should just be shifted to right side so that the area is covered by all the lines...
2 square boxes and two triangles must be covered by all the lines....
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
1
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manpreetsingh86 wrote:
sunny3011 wrote:
The area bounded by the three curves |x + y | = 1, | x | = 1, and | y | = 1, is equal to
a. 1
b. 2
c. 3
d. 4
e. 8

I have tried to plot graph of lines and I got area from |x|=1 and |y|=1. but what about line |x+y|=1???
Confused and need to clarify the concepts?

Source: GMAT Tutor Question bank

|x+y|=1 can be written as x+y=1 or x+y=-1. These are two straight lines, which are mirror image of each as shown in the following graph.

|x|=1 and |y|=1 are also drawn on the graph with red and blue lines respectively. The required area is shown in the grey blocks. which is equal to 3(2+1/2+1/2). (2 block of area 1 and two triangles of area =1/2)

The red line should be 1 square to the right? looks like it represents x=-2 instead of x=-1
Please correct me if Im wrong
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
Bunuel - We get x+y= -1 and x+y= 1. How do I plot the given equation |x + y | = 1 on the graph?
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
1
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area = 6*0.5*1*1 = 3

are the shaded regions bounded by these curves correct?
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region.jpg [ 1.48 MiB | Viewed 11215 times ]

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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
sunny3011 wrote:
The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = 1, is equal to

A. 1
B. 2
C. 3
D. 4
E. 8

I have tried to plot graph of lines and I got area from |x|=1 and |y|=1. but what about line |x+y|=1???
Confused and need to clarify the concepts?

Source: GMAT Tutor Question bank

See the attached figure.

Now |x|=1 will give two straight lines perpendicular to x-axis crossing x at 1 and the other at -1.
Similarly |y|=1 will give two straight lines perpendicular to y-axis crossing y at 1 and the other at -1.

Thus |x|=1 and |y|=1 enclose a square of side 1-(-1)=2, so area of square = 2*2=4. (also shown by blue colour in graph).
So our area has to be less than or at the max equal to 4
. Eliminate E.

Draw lines |x+y|=1. The area which it excludes from 4 is shown by the star.
Now each square in the graph is 1*1, and |x|=1 and |y|=1 includes 4 such squares of area 1 each.=>184=1
But each of the $$\triangle$$ shown by the two stars are equal to half of the square 1*1. Thus, area of two triangles combined is equal to that of one square of 1*1 or 1.
Area, therefore, covered by |x|=1, |y|=1, and |x+y|=1 is 4-1=3.

Also since each of the |x|=1, |y|=1, and |x+y|=1 are linear equations and therefore equation of lines, it would be appropriate to mention them as lines rather than curve.
Attachments

Untitleda.png [ 7.51 KiB | Viewed 11030 times ]

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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
preetamsaha wrote:
area = 6*0.5*1*1 = 3

are the shaded regions bounded by these curves correct?

The region is the unshaded portion in middle. The area has to be enclosed by all the three lines.
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
preetamsaha wrote:
area = 6*0.5*1*1 = 3

are the shaded regions bounded by these curves correct?

chetan2u
in my picture, all the areas are enclosed by three lines. am i wrong? although in your approach I found it very much logical.
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
preetamsaha wrote:
preetamsaha wrote:
area = 6*0.5*1*1 = 3

are the shaded regions bounded by these curves correct?

chetan2u
in my picture, all the areas are enclosed by three lines. am i wrong? although in your approach I found it very much logical.

No, when we say area enclosed by |x|=1, all the area above the line x=1 and below the area x=-1 will be discarded.
So all the ares you have chosen is actually out of at least one of the regions enclosed by |x|=1, or by |y|=1 or by |x+y|=1.

Most of the time, when you talk of 'enclosed by lines' that are on either side of ORIGIN, the area that includes the origin is the correct portion.
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
chetan2u
Okay. Thanks a lot.
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
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Re: The area bounded by the three curves |x + y | = 1, |x| = 1, and |y| = [#permalink]
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