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Intern  Joined: 21 Apr 2010
Posts: 3
The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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9
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Difficulty:   65% (hard)

Question Stats: 59% (01:57) correct 41% (01:51) wrong based on 861 sessions

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The area bounded by the curves |x + y| = 1 and |x - y| = 1 is

A. 3
B. 4
C. 2
D. 1
E. None
Math Expert V
Joined: 02 Sep 2009
Posts: 58037
Re: area bounded by the curves  [#permalink]

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9
15
fortsill wrote:
fivezero7 wrote:
anni wrote:
thank you for the reply, is it required to plot these lines and then calculate?
can we get directly i mean is there any formula to solve this problem?
thanks

hi anni,
there is no need to plot it, once you have mastered the art of visualization.
the way i did it is as under.
take x+y=1
it intersects the axes at (1,0) and (0,1) and makes a right angled triangle with the axes with each side (other than the hypotenuse) as 1
hence the area of this triangle is 0.5*1*1 = 0.5 sq. units.

all other lines are symmetrical and form 3 more congruent triangles with the axes at different points.
so the total area will be 0.5+0.5+0.5+0.5 = 2 sq units.
hope i am clear Very clear, and amazed to see how much was packed under that question.
But, am curious if there's any other way to solve the problem?

The area bounded by the curves |x + y| = 1, |x - y| = 1 is
A. 3
B. 4
C. 2
D. 1
E. None

|x+y|=1 represents two lines: x+y=1 and x+y=-1 --> y=1-x and y=-1-x. Find the x and y intercept of these lines to plot;
|x-y|=1 represents two lines: x-y=1 and x-y=-1 --> y=x-1 and y=x+1. Find the x and y intercept of these lines to plot;

Notice that these lines are mirror images of each other. Here is a square you get when you plot them:
Attachment: Area.gif [ 4.93 KiB | Viewed 45921 times ]
Notice that the diagonal of this square is equal to 2 (the difference between x intercepts). Area of a square is diagonal^2/2=2^2/2=2.

Check similar questions to practice:
m06-5-absolute-value-108191.html
graphs-modulus-help-86549.html
m06-q5-72817.html
if-equation-encloses-a-certain-region-110053.html
area-of-region-126117.html

Hope it helps.
_________________
Intern  Joined: 08 Apr 2010
Posts: 27
Re: area bounded by the curves  [#permalink]

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3
2
anni wrote:
The area bounded by the curves $$|x + y| = 1,$$ $$|x - y| = 1$$ is

A.3
B.4
C.2
D.1
E. None

please, help how to solve ?

hi,
these equations represent four different equations
x+y=1
x+y=-1
x-y=1
x-y=-1
once you plot these on the graph, you will easily find the area under these equations.

just for the record, the answer shall be 2.
_________________
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fivezero7
##### General Discussion
Intern  Joined: 21 Apr 2010
Posts: 3
Re: area bounded by the curves  [#permalink]

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fivezero7 wrote:
anni wrote:
The area bounded by the curves $$|x + y| = 1,$$ $$|x - y| = 1$$ is

A.3
B.4
C.2
D.1
E. None

please, help how to solve ?

hi,
these equations represent four different equations
x+y=1
x+y=-1
x-y=1
x-y=-1
once you plot these on the graph, you will easily find the area under these equations.

just for the record, the answer shall be 2.

thank you for the reply, is it required to plot these lines and then calculate?
can we get directly i mean is there any formula to solve this problem?
thanks
Intern  Joined: 08 Apr 2010
Posts: 27
Re: area bounded by the curves  [#permalink]

### Show Tags

3
2
anni wrote:
thank you for the reply, is it required to plot these lines and then calculate?
can we get directly i mean is there any formula to solve this problem?
thanks

hi anni,
there is no need to plot it, once you have mastered the art of visualization.
the way i did it is as under.
take x+y=1
it intersects the axes at (1,0) and (0,1) and makes a right angled triangle with the axes with each side (other than the hypotenuse) as 1
hence the area of this triangle is 0.5*1*1 = 0.5 sq. units.

all other lines are symmetrical and form 3 more congruent triangles with the axes at different points.
so the total area will be 0.5+0.5+0.5+0.5 = 2 sq units.
hope i am clear _________________
CHEERS
fivezero7
Intern  Joined: 24 Feb 2012
Posts: 29
Re: area bounded by the curves  [#permalink]

### Show Tags

3
fivezero7 wrote:
anni wrote:
thank you for the reply, is it required to plot these lines and then calculate?
can we get directly i mean is there any formula to solve this problem?
thanks

hi anni,
there is no need to plot it, once you have mastered the art of visualization.
the way i did it is as under.
take x+y=1
it intersects the axes at (1,0) and (0,1) and makes a right angled triangle with the axes with each side (other than the hypotenuse) as 1
hence the area of this triangle is 0.5*1*1 = 0.5 sq. units.

all other lines are symmetrical and form 3 more congruent triangles with the axes at different points.
so the total area will be 0.5+0.5+0.5+0.5 = 2 sq units.
hope i am clear Very clear, and amazed to see how much was packed under that question.
But, am curious if there's any other way to solve the problem?
Intern  Joined: 12 Mar 2012
Posts: 9
Re: The area bounded by the curves |x + y| = 1, |x - y| = 1 is  [#permalink]

### Show Tags

anni wrote:
The area bounded by the curves |x + y| = 1, |x - y| = 1 is

A. 3
B. 4
C. 2
D. 1
E. None

please, help how to solve ?

One more method:

solve following equns:

X+Y=1
-X-y=1
X-y=1
-X+y=1

Out put X= +_ 1
Y=+_1

Plot the values of X and Y on graph, you will see the square

Now use pythagoras thm to find diagonal,which will be the side of that square.

= Sqaure root 2

square it and ans will be 2
Manager  Joined: 27 Dec 2011
Posts: 53
Re: The area bounded by the curves |x + y| = 1, |x - y| = 1 is  [#permalink]

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the question threw me away when it said "curves" and I started thinking about parabola, It should have said "lines" instead.

@Bunuel,
Does gmat confuses us with this kind of language?

thanks!
Senior Manager  Joined: 13 Aug 2012
Posts: 405
Concentration: Marketing, Finance
GPA: 3.23
Re: The area bounded by the curves |x + y| = 1, |x - y| = 1 is  [#permalink]

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1
2
(1) Derive all equations

|x + y| = 1
eq1: x + y = 1
eq2: x + y = -1

|x - y| = 1
eq3: x - y = 1
eq4: y - x = 1

(2) Plot your graph using x=0 and y=0.

eq1: 0,1 and 1,0
eq2: 0,-1 and -1,0
eq3: 0-1 and 1,0
eq4: 0,-1 and -1,0

(3) You will recognize a region that is a square with a diagonal of 2
(4) Calculate the area.

diagonal = side * $$\sqrt{2}$$
side = $$\frac{2}{\sqrt{2}}$$
side = $$\sqrt{2}$$

Area = $$side^2$$ = $$\sqrt{2}^2$$ = $$2$$

For detailed solutions for other similar problems. http://burnoutorbreathe.blogspot.com/2012/12/absolute-values-solving-for-area-of.html
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Math Expert V
Joined: 02 Sep 2009
Posts: 58037
Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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anni wrote:
The area bounded by the curves |x + y| = 1 and |x - y| = 1 is

A. 3
B. 4
C. 2
D. 1
E. None

The question seems confusing at first since it said the curves. But if you know how to deal with absolute values, you can come up with the solution pretty quickly.

C
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Intern  Joined: 03 Jul 2015
Posts: 30
Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

why only (0,1) (1,0) should only pick? for x+y=1, if i chose (10,-9) then it still become 1, can you please explain me what exactly the method of chosing number while finding area for this type of math
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The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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Hi Bunuel, just a question, I am getting confused with this. Can't we just find all the solutions for x and y by using just one equation?
If we take the first equation:
|x + y| = 1
and say that y = 0 then x = 1 or -1
and if x = 0 then y =1 and -1

Would that be incorrect? And if that's the case, what is the difference between this equation |x + y| = 1 and this one |x|+|y|=1
Thanks a lot
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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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2
anni wrote:
The area bounded by the curves |x + y| = 1 and |x - y| = 1 is

A. 3
B. 4
C. 2
D. 1
E. None

x+y=1 -> y=-x+1 -> slope -1.
x+y=-1 -> y=-x-1 -> slope -1. we have 2 parallel lines.

x-y=1 -> y=x-1 => slope 1.
x-y=-1 -> y=x+1 -> slope 1. -> we have another 2 parallel lines.

i simply drew the lines, and for the sake of getting the image, try x=1 then y=1 for all the equations.
i got a square like shape, with the points of intersection at (-1; 0); (0; 1); (1; 0); (0; -1)
since the diagonal of the square is the x axis, and it has a length of 2, we can apply the 45-45-90 triangle rule, and see that 2=x*sqrt(2), where x is the side of the square.
the side of the square is sqrt(2). now, we need to find the area -> sqrt(2) squared is equal to 2.
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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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anik19890 wrote:
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

why only (0,1) (1,0) should only pick? for x+y=1, if i chose (10,-9) then it still become 1, can you please explain me what exactly the method of chosing number while finding area for this type of math

you would find more points on THE LINE, but we are asked for the area of the figure when the 4 lines intersect.
the 4 points of intersection are as shown in the figure in bunuel's post.
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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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Excellent question...
Tested understanding of Inequalities, Geometry and visualization...
Is it a Q 51 level question?
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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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I have a query with the wording of the question
it says : The area bounded by the curves |x + y| = 1, |x - y| = 1 is
A. 3
B. 4
C. 2
D. 1
E. None

Based on answers here it appears we consider them as four straight lines (not curves). Curves usually are written by x^2 + y^2 format; but since it said curves I thought it would be circular--> so in case its a non - square in the equation;I assume it to be lines not curves?? Will keep in mind in case I come across such language/ format later
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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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An easy question instead <700 probably

|x+y|=1 , gives 2 equations x+y = + 1 and x+y =-1
|x-y| =1 gives x-y=1 and x-y = -1
we know they will form an enclosed figure on x and y axis.
so directly find the points = put x=0 and y=0
we get (0,1) (0,-1) (1,0) (-1,0)
when u actually plot the distance between opposite points are = 2 on each diogonal which cut at 90 degrees at O , origin
there fore

A = 0.5 D1D2 = 0.5x2x2 = 2 which is C
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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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anni wrote:
The area bounded by the curves |x + y| = 1 and |x - y| = 1 is

A. 3
B. 4
C. 2
D. 1
E. None

The total Area = $$4*\frac{1}{2}*1*1$$
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Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is  [#permalink]

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_________________ Re: The area bounded by the curves |x + y| = 1 and|x - y| = 1 is   [#permalink] 31 Mar 2019, 02:51
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