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

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.

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

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

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

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