What is the number of solutions of x = |x-|30-2x||?

Question

What is the number of solutions of  x = |x-|30-2x|| ?

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

GMATGuruNY · Accepted Answer

| x-|30-2x| | = x if the blue portion is equal to x or -x.

Case 1: x-|30-2x| = x 
0 = |30-2x|
The equation above is valid if 30-2x=0:
30-2x = 0
30 = 2x
15 = x

Case 2: x-|30-2x| = -x 
2x = |30-2x|

Case 2a:
2x = 30-2x
4x = 30
x = 30/4 = 7.5

Case 2b:
-2x = 30-2x
0 = 30
Not valid.

Two solutions:
x=15 and x=7.5

C

Archit3110 · Answer

solve for expression ;  x = |x-|30-2x|| 
we get x= 15 and 15/2
total solutions ; 2
IMO C

Kinshook · Answer

Since x = |y| => x>0 Let us divide x>0 in 2 regions Region 1 : 0x>10 x = 3x -30 x=15 Solution 1. Now suppose 015 x=|x-(2x-30)| x= |30 -x| Suppose x>30 x = x-30 No Solution Now suppose 15

Kinshook · Answer

Can this question be solved without dividing number line into seperate regions?
If yes, please provide way to solve such questions.

nick1816 · Answer

Case 1 if x=<10, x<|30-2x| and 30-2x>0 x=30-2x-x x=7.5 Case 2 if 10|30-2x| and 30-2x>0 x=x-30+2x x=15 Case 3 if 15|30-2x| and 30-2x<0 x= x- 2x+30 x= 15 (not possible) Case 4 if x>30, x<|30-2x| and 30-2x<0 x=2x-30-x Not possible There are 2 values of x possible

Kinshook · Answer

How do you decide exact regions in a nested modulus equation without opening some inner modulus first?
e.g In x<10, how do you identify 10?

nick1816 · Answer

I didn't explicitly mention it. You gotta do couple more steps to divide the regions. When x>|30-2x| i) for x>15 x>2x-30 x<30 ii) for x<15 x>30-2x x>10 Similarly you can find other cases.

neelgmat · Answer

hi.. can you please elaborate how you directly decided the the values of the two regions??

Kinshook · Answer

I tried to open |2x-30| first

Posted from my mobile device

MathRevolution · Answer

=>

The equation  x = |x-|30-2x||  is equivalent to x = |x-2|x-15||

If  x ≥ 15 , then  x = |x-2|x-15||  or  x = | x – 2(x-15) | = | x – 2x + 30 | = | -x + 30 | = | x – 30 |

If  x ≥ 30 , then  x = | x – 30 | = x – 30  or  0 = -30 , which doesn’t make sense.

If  15 ≤ x < 30,  then  x = - ( x – 30 ) = -x + 30  or  2x = 30.  It follows that  x = 15.

If  x < 15 , then  x = |x-2|x-15||  is equivalent to  x = | x + 2(x-15) | = | x + 2x - 30 | = | 3x - 30 | = 3| x – 10 |

If  10 ≤ x < 15 , then  x = 3| x – 10 | = 3(x-10) = 3x -30,  so,  2x – 30 = 0.  It follows that  x = 15,  which is not a solution since  10 ≤ x < 15.

If  x < 10,  then  x = 3| x – 10 | = -3(x-10) = -3x + 30  and  4x = 30.

So,  x = 7.5.

Thus, there are two solutions:  7.5  and  15.

Therefore, the answer is C.
Answer: C

Gaurav015 · Answer

I took |30-2x| = a 
then x = |x-a|
So we will get a = 0. 
Putting the value of a here, we will get x = 15 
again opening with the negative sign X = -x + a

We will get 2X = a 
|30-2x| = 2X, then x = 7.5
We get the value of X as  15 and 7.5.  
Hence,  option C

What is the number of solutions of x = |x-|30-2x||?

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What is the number of solutions of x = |x-|30-2x||?

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