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

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by Max@Math Revolution » Wed Jul 03, 2019 11:58 pm

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C

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[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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by GMATGuruNY » Thu Jul 04, 2019 4:00 am
Max@Math Revolution 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
|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

The correct answer is C.
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by Max@Math Revolution » Sun Jul 07, 2019 5:08 pm
=>

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
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