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

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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
MathRevolution wrote:
[GMAT math practice question]

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

$$A. 0$$

$$B. 1$$

$$C. 2$$

$$D. 3$$

$$E. 4$$

Since x = |y| => x>0

Let us divide x>0 in 2 regions

Region 1 : 0<x<15
x = |x- (30-2x|
x= |3x - 30| = 3 |x-10|
Suppose 15>x>10
x = 3x -30
x=15 Solution 1.
Now suppose 0<x<10
x = 30 -3x
4x = 30
x = 7.5 Solution 2

Region 2: x>15
x=|x-(2x-30)|
x= |30 -x|
Suppose x>30
x = x-30 No Solution
Now suppose 15<x<30
x = 30 -x
x=15 Which is same as Solution 1.

We see that there are only 2 solutions.

IMO C
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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
Archit3110 wrote:
solve for expression ; $$x = |x-|30-2x||$$
we get x= 15 and 15/2
total solutions ; 2
IMO C

MathRevolution wrote:
[GMAT math practice question]

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

$$A. 0$$

$$B. 1$$

$$C. 2$$

$$D. 3$$

$$E. 4$$

Can this question be solved without dividing number line into seperate regions?
If yes, please provide way to solve such questions.
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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
Case 1
if x=<10, x<|30-2x| and 30-2x>0

x=30-2x-x
x=7.5

Case 2
if 10<x=<15, x>|30-2x| and 30-2x>0

x=x-30+2x
x=15

Case 3
if 15<x=<30, x>|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

MathRevolution wrote:
[GMAT math practice question]

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

$$A. 0$$

$$B. 1$$

$$C. 2$$

$$D. 3$$

$$E. 4$$
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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
nick1816 wrote:
Case 1
if x=<10, x<|30-2x| and 30-2x>0

x=30-2x-x
x=7.5

Case 2
if 10<x=<15, x>|30-2x| and 30-2x>0

x=x-30+2x
x=15

Case 3
if 15<x=<30, x>|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

MathRevolution wrote:
[GMAT math practice question]

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

$$A. 0$$

$$B. 1$$

$$C. 2$$

$$D. 3$$

$$E. 4$$

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?
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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
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.

Kinshook wrote:
nick1816 wrote:
Case 1
if x=<10, x<|30-2x| and 30-2x>0

x=30-2x-x
x=7.5

Case 2
if 10<x=<15, x>|30-2x| and 30-2x>0

x=x-30+2x
x=15

Case 3
if 15<x=<30, x>|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

MathRevolution wrote:
[GMAT math practice question]

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

$$A. 0$$

$$B. 1$$

$$C. 2$$

$$D. 3$$

$$E. 4$$

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?
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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
Kinshook wrote:
MathRevolution wrote:
[GMAT math practice question]

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

$$A. 0$$

$$B. 1$$

$$C. 2$$

$$D. 3$$

$$E. 4$$

Since x = |y| => x>0

Let us divide x>0 in 2 regions

Region 1 : 0<x<15
x = |x- (30-2x|
x= |3x - 30| = 3 |x-10|
Suppose 15>x>10
x = 3x -30
x=15 Solution 1.
Now suppose 0<x<10
x = 30 -3x
4x = 30
x = 7.5 Solution 2

Region 2: x>15
x=|x-(2x-30)|
x= |30 -x|
Suppose x>30
x = x-30 No Solution
Now suppose 15<x<30
x = 30 -x
x=15 Which is same as Solution 1.

We see that there are only 2 solutions.

IMO C

hi.. can you please elaborate how you directly decided the the values of the two regions??
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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
neelgmat wrote:
Kinshook wrote:
MathRevolution wrote:
[GMAT math practice question]

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

$$A. 0$$

$$B. 1$$

$$C. 2$$

$$D. 3$$

$$E. 4$$

Since x = |y| => x>0

Let us divide x>0 in 2 regions

Region 1 : 0<x<15
x = |x- (30-2x|
x= |3x - 30| = 3 |x-10|
Suppose 15>x>10
x = 3x -30
x=15 Solution 1.
Now suppose 0<x<10
x = 30 -3x
4x = 30
x = 7.5 Solution 2

Region 2: x>15
x=|x-(2x-30)|
x= |30 -x|
Suppose x>30
x = x-30 No Solution
Now suppose 15<x<30
x = 30 -x
x=15 Which is same as Solution 1.

We see that there are only 2 solutions.

IMO C

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

I tried to open |2x-30| first

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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
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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 $$15.$$

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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
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Re: What is the number of solutions of x = |x-|30-2x||? [#permalink]
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