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.
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