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Intern  S
Status: Fortune favours the brave
Joined: 17 Jul 2019
Posts: 9
Location: Russian Federation
GMAT 1: 710 Q49 V39 GPA: 3.98
If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 21% (02:07) correct 79% (02:23) wrong based on 75 sessions

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If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions Kudos are always appreciated Director  D
Joined: 19 Oct 2018
Posts: 961
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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2
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions Kudos are always appreciated Attachments 1111.PNG [ 18.44 KiB | Viewed 1075 times ]

Intern  S
Status: Fortune favours the brave
Joined: 17 Jul 2019
Posts: 9
Location: Russian Federation
GMAT 1: 710 Q49 V39 GPA: 3.98
If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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3
Such tasks are easy to solve graphically: solutions to the system of equation are points of intersection of graphs of each equation.

Remember: When a module sign is "layed" on F(x), graphical part of F(x), which lays below the X-axis, is reflected over the X-axis. This operation gives you the graph of |F(x)|.

Thus, the given task is solved with consequent graphics: As already discussed, to obtain ||x|-0.34| graph, we reflect "negative" part of |x|-0.34 over X-axis. Then shift ||x|-0.34| by 0.66 upwards to obtain ||x|-0.34|+0.66. Than we draw the plot of x^2+y^2=1

Finally, one can see 3 points of intersection -> 3 solutions to the system -> x can take 3 values -> answer is D
Intern  Joined: 22 Aug 2019
Posts: 1
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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This question is a tough one Manager  S
Joined: 17 Jul 2017
Posts: 191
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions Kudos are always appreciated nick1816
Bunuel
What is wrong with the given solution in the image:
here x=0 or 1 0r -1

but i consider case last one in pic (from d)
here i get x=.84 and y =-.52
then also x^2+y^2=1 then y are we nopt considering such cases
Attachments IMG20190827210630.jpg [ 3.48 MiB | Viewed 1004 times ] IMG20190827210624.jpg [ 3.52 MiB | Viewed 1007 times ]

Director  D
Joined: 19 Oct 2018
Posts: 961
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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There are couple of problems with this approach for this particular question.
1. This approach can take a lot of time. Whenever you see second order equations, go for the graphical approach.

2. You didn't find the domain for x, for each equation.
For example-
||x|-0.34|+0.66=y
y=x-0.34+0.66, when x≥0.34
y=0.34-x+0.66, when $$0.34 ≥x ≥0$$
y=0.34+x+0.66, when 0≥x≥-0.34
y=0.34-x+0.66, when x≤-0.34

Now you can solve each case, and look whether the solution lies in the domain or not.

vanam52923 wrote:
Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions Kudos are always appreciated nick1816
Bunuel
What is wrong with the given solution in the image:
here x=0 or 1 0r -1

but i consider case last one in pic (from d)
here i get x=.84 and y =-.52
then also x^2+y^2=1 then y are we nopt considering such cases
Manager  S
Joined: 17 Jul 2017
Posts: 191
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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nick1816 wrote:
There are couple of problems with this approach for this particular question.
1. This approach can take a lot of time. Whenever you see second order equations, go for the graphical approach.

2. You didn't find the domain for x, for each equation.
For example-
||x|-0.34|+0.66=y
y=x-0.34+0.66, when x≥0.34
y=0.34-x+0.66, when $$0.34 ≥x ≥0$$
y=0.34+x+0.66, when 0≥x≥-0.34
y=0.34-x+0.66, when x≤-0.34

Now you can solve each case, and look whether the solution lies in the domain or not.

vanam52923 wrote:
Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions Kudos are always appreciated nick1816
Bunuel
What is wrong with the given solution in the image:
here x=0 or 1 0r -1

but i consider case last one in pic (from d)
here i get x=.84 and y =-.52
then also x^2+y^2=1 then y are we nopt considering such cases

My bad ,thanks a lot But are such questions asked on GMAT?
SVP  P
Joined: 03 Jun 2019
Posts: 1689
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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nick1816 VeritasKarishma Bunuel chondro48 Is there an algebra way to solve this question?

nick1816 wrote:
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions Kudos are always appreciated _________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

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Senior Manager  P
Joined: 30 Sep 2017
Posts: 379
Concentration: Technology, Entrepreneurship
GMAT 1: 720 Q49 V40 GPA: 3.8
WE: Engineering (Real Estate)
If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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Kinshook wrote:
nick1816 VeritasKarishma Bunuel chondro48 Is there an algebra way to solve this question?

nick1816 wrote:
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

Kudos are always appreciated Hi Kinshook nick1816 VeritasKarishma and Bunuel,

Yes, there is. Below is a framework to do so. Correct me, experts, if I make mistake.

Note that using algebra, the solution is laborious and time consuming.

Transform $$x^2=1-y^2$$ to |x|=√(1-y^2). Substituting |x| with √(1-y^2) into the second equation, we get:
|√(1-y^2)-0.34|=y-0.66.

If √(1-y^2) > 0.34, then √(1-y^2)-0.34 = y-0.66 --> √(1-y^2)= y-0.34. After squaring both sides and performing tedious calculation, we will find that there are two solution points available when √(1-y^2) > 0.34.

If √(1-y^2) <= 0.34, then 0.34-√(1-y^2) = y-0.66 --> √(1-y^2)=1-y. It is obvious that y=1 satisfies the equation and correspondingly |x|=√(1-y^2)=0. Thus, there is one solution point (0,1) when √(1-y^2) <= 0.34.

Finally, there are 3 values that x can take.

Final answer is (D)

Originally posted by chondro48 on 27 Aug 2019, 17:04.
Last edited by chondro48 on 28 Aug 2019, 00:39, edited 1 time in total.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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Kinshook wrote:
nick1816 VeritasKarishma Bunuel chondro48 Is there an algebra way to solve this question?

nick1816 wrote:
1 Kudos for an excellent question.
IMO graphical approach can save lot of time in this question.

We just need to figure out whether point of minima of ||x|-0.34|+0.66=y lies within the circle or not, and whether local maxima at x=0 lies within circle or not.
$$x^2+y^2=1$$ is a curve of a circle of radius 1.

At x=0, ||x|-0.34|+0.66= 1, which lies on the circle.

Minima of $$||x|-0.34|+0.66=y$$ occurs when |x|-0.34=0 or x=+0.34 and -0.34, and y=0.66, lie inside the circle.

Now we are good to draw the curve.

There are 3 points where both curves meet.

D

Philipp98 wrote:
If $$x^2+y^2=1$$ and $$||x|-0.34|+0.66=y$$, than how many values can $$x$$ take?

A. 0
B. 1
C. 2
D. 3
E. 4

______
Smth similar I met in my GMAT exam, thus make sure you are familiar with such questions Kudos are always appreciated You can solve it using the usual method of definition of absolute values (as done by others here) but if you are unable to solve it graphically, it is probably a good idea to skip the question. With algebra, it will be far too time consuming and error prone.
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Intern  B
Joined: 26 Oct 2010
Posts: 14
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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nick1816

The explanation given by you to find out the value of y at the equation's minima is crystal clear. We have arrived at y = 0.66 at minima i.e x = +/- 0.34.

Now, how do we figure if y = 0.66 falls inside the circle?

Ofcourse we can substitute x = 0.34 in the circle equation to find out the value of y. And if the value of y > 0.66, we can deduce that 0.66 is inside the circle. However, this is a time consuming process that requires squaring and square-roots

Is there are short cut to find if 0.66 falls inside the circle?

Director  D
Joined: 19 Oct 2018
Posts: 961
Location: India
Re: If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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1
0.34≃1/3
0.66≃2/3
$$(\frac{1}{3})^2+(\frac{2}{3})^2$$= $$\frac{1}{9}+\frac{4}{9}$$=$$\frac{5}{9}$$<1

GMATaspirant641 wrote:
nick1816

The explanation given by you to find out the value of y at the equation's minima is crystal clear. We have arrived at y = 0.66 at minima i.e x = +/- 0.34.

Now, how do we figure if y = 0.66 falls inside the circle?

Ofcourse we can substitute x = 0.34 in the circle equation to find out the value of y. And if the value of y > 0.66, we can deduce that 0.66 is inside the circle. However, this is a time consuming process that requires squaring and square-roots

Is there are short cut to find if 0.66 falls inside the circle?

Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?  [#permalink]

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GMATaspirant641 wrote:
nick1816

The explanation given by you to find out the value of y at the equation's minima is crystal clear. We have arrived at y = 0.66 at minima i.e x = +/- 0.34.

Now, how do we figure if y = 0.66 falls inside the circle?

Ofcourse we can substitute x = 0.34 in the circle equation to find out the value of y. And if the value of y > 0.66, we can deduce that 0.66 is inside the circle. However, this is a time consuming process that requires squaring and square-roots

Is there are short cut to find if 0.66 falls inside the circle?

Check the solution given by Philipp98 above. Notice how the graph of ||x| - 0.34| + 0.66 = y is drawn.
Also note that each green line has slope 1 or -1. Since the graph intersects the circle at (0, 1), the line of slope 1 will be inside the circle (45 degrees line). Hence you know that it will intersect the circle in 3 points.

Check out these two posts on how to draw absolute value graphs:

https://www.veritasprep.com/blog/2011/0 ... h-to-mods/
https://www.veritasprep.com/blog/2014/1 ... solutions/
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Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options > If x^2+y^2=1 and ||x|-0.34|+0.66=y, how many values can x take?   [#permalink] 29 Aug 2019, 23:13
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