M37-85

Question

What is the median of all the values of  x , which satisfy  ||x – 3| – 5| = 3  ?

A.  1 
B.  2 
C.  3 
D.  5 
E.  6

Bunuel · Answer

Official Solution:

What is the median of all the values of  x , which satisfy  ||x – 3| – 5| = 3  ?

A.  1 
B.  2 
C.  3 
D.  5 
E.  6

||x – 3| – 5| = 3  means that  |x – 3| – 5 = 3  or  |x – 3| – 5 = -3 .
 
 CASE 1: 
 
 |x – 3| – 5 = 3 ;
 
 |x – 3| = 8 ;
 
 x=11  or  x=-5 
 
 CASE 2: 
 
 |x – 3| – 5 = -3 ;
 
 |x – 3| = 2 ;
 
 x=5  or  x=1 
 
So,  x  can be (in ascending order): -5, 1, 5, or 11. The median of the values is  \frac{1+5}{2}=3 .

Answer: C

MBAToronto2024 · Answer

KarishmaB  I tried to square both sides and get the result. I failed to get all values. What's wrong with this approach?

KarishmaB · Answer

Here we have double mod. How can we square both sides to get the answer?

||x–3|–5|=3

^2 = 9

|x-3|^2 + 25 - 2*5 * |x - 3| = 9

You get x^2 term, x term and there is a |x-3| term. What do you do next?

This question can be done using the graphical approach or using algebra as done by Bunuel above.

aronbhati · Answer

Hi Bunuel, here we will satisfy the all values of X by considering X>5 & X<5 or X>3 &X<3

Bunuel · Answer

You could split into cases with x < 3 and x ≥ 3, leading to |x - 8| = 3 and |x + 2| = 3, and then further split into x > 8 and x ≤ 8, and x > -2 and x ≤ -2. However, the method shown in the solution is much faster and more efficient.

padmru · Answer

I don’t quite agree with the solution. I did not understand why its solve differently than most other similar style problems. For instance, absolute value of first bracket is 3 or -3. 
But then the absolute value of second bracket is considered by not adjusting -5 to +ve value of inner absolute value. x-8 = + or - 3.

Could some one explain why absolute of absolute value Is considered differently.

Bunuel · Answer

Not sure that I follow what you mean, but ||x - 3| - 5| = 3 splits as |x - 3| - 5 = 3 or |x - 3| - 5 = -3, so |x - 3| = 8 or |x - 3| = 2. What is unclear here? Also, you don’t agree with this solution or didn’t understand it? Anyway, here is a thread with alternative solutions: https://gmatclub.com/forum/hot-competit ... 33254.html Hope it helps.

M37-85

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