Suppose that |x + y| + |x -y| = 2. What is the maximum possible value

Question

Suppose that |x + y| + |x -y| = 2. What is the maximum possible value of x^2 - 6x + y^2?

A. 5
B. 6
C. 7
D. 8
E. 9

eswarchethu135 · Accepted Answer

if we open mod there are 4 cases possible

Case 1) Both are positive

x + y + x - y = 2
2x = 2
x = 1

Case 2) |x + y| is negative

-x - y + x - y = 2
-2y = 2
y = -1

Case 3) |x - y| is negative
x + y - x + y = 2
2y = 2
y = 1

Case 4) Both are negative
- x - y - x + y = 2
-2x = 2
x = -1

So from the above 4 cases, we can say  x = \pm{1}  and  y = \pm{1}

Substituting in the eq:  x^2 - 6x + y^2

1 - 6(\pm{1}) + 1

This value will be maximum when x = -1

= 1 -6(-1) + 1

=8

OPTION:  D

chetan2u · Answer

We can find our solution by logic.
(I) Both the terms |x + y| and |x -y| are at least 0, and cannot be more than 2 as their sum itself is given as 2.
(II) If you have to maximize  x^2 - 6x + y^2 , we will try to maximize -6x as that will be the largest value, so x should be negative.

Now |x| and |y| should not be greater than 1, so both can be 1..
x has to be negative, so x=-1, and y =1..
|x + y| + |x -y| =|-1+1|+|-1-1|= 2...Yes

Value of  x^2 - 6x + y^2  will become  (-1)^2 - 6(-1) + (-1)^2=1+6+1=8

D

gvij2017 · Answer

How I approached this question is as follows. 
I know that every term in equation is positive. Therefore, I squared both sides. 
And got (x^2 + y^2)= 2.

Now I have maximize the x^2 + y^2 - 6x
To maximize 2-6x, x must be negative and can be -1. 
So, maximum value is 8.

I think my approach to such questions in which both sides are positive is correct.

foryearss · Answer

Hi , I'm curious , when you square both sides , you got 2 *|x + y| * |x -y| in your way , how you did deal with that?

vinayakvaish · Answer

(|x+y| + |x-y|) ^2 = 4
2(x^2 + y^2) + 2( x^2-y^2)=4
4x^2=4
x=+/- 1

to maximize, x must be -1

put this in original equation, y can be 0 or 1

since we are looking to maximize, choose 1

Answer= 1+6+1= 8

drasticgre · Answer

Hei vinayakvaish
your 2nd line - isnt it possible 2(x^2 + y^2)  - 2( x^2-y^2) =4 ?

hadimadi · Answer

Hi,

drasticgre ,  vinayakvaish

Let a=root((x+y)^2), b=root((x-y)^2)

We have:

a+b=2, now we square to get
(a+b)^2=4
a^2+2ab+b^2=4, plugging in our initial terms we get
x^2+2xy+y^2+2(lx+yl*lx-yl)+x^2-2xy+y^2=4
2x^2+2y^2+2*lx^2-y^2l=4
x^2+y^2+lx^2-y^2l=2

C1, lxl>lyl

2x^2=2
-> x_1,2=1, -1

x has to be the negative solution, because the term x^2-6x will then be greatest. The corresponding y is then -1 or 1, and we get 8
for our equation for both y's

C2, lxl<lyl

2y^2=2
-> y_1,2=1, -1

We again need a negative x, so in total, we have 8 again

Bunuel ,  nick1816 , could you kindly check if this solution is correct? I feel like something is wrong or it's too lengthy

Oppenheimer1945 · Answer

Max((x-3)^2+y^2-9) = max dist of (3,0) from a point in square=4^2+1^2-9=8

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

Suppose that |x + y| + |x -y| = 2. What is the maximum possible value

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Suppose that |x + y| + |x -y| = 2. What is the maximum possible value

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards