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How many solutions has the equation ||x-3|-2|=1?

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by Max@Math Revolution » Mon Jan 22, 2018 12:01 am
[GMAT math practice question]

How many solutions has the equation ||x-3|-2|=1?

A. 0
B. 1
C. 2
D. 3
E. 4
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by GMATGuruNY » Mon Jan 22, 2018 4:09 am
Max@Math Revolution wrote:[GMAT math practice question]

How many solutions has the equation ||x-3|-2|=1?

A. 0
B. 1
C. 2
D. 3
E. 4
Case 1: |x-3|-2 = 1
|x-3| = 3
If x-3=3, then x=6.
If x-3=-3, then x=0.

Case 2: |x-3|-2 = -1
|x-3| = 1
If x-3=1, then x=4.
If x-3=-1, then x=2.

The equation has the four solutions in blue: 6, 0, 4 and 2.

The correct answer is E.
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by Jeff@TargetTestPrep » Tue Jan 23, 2018 9:10 am
Max@Math Revolution wrote:[GMAT math practice question]

How many solutions has the equation ||x-3|-2|=1?

A. 0
B. 1
C. 2
D. 3
E. 4
We see that |x-3| - 2 could be 1 or -1 since both |1| and |-1| are equal to 1.

1) If |x - 3| - 2 = 1, then |x - 3| = 3. So x - 3 could be 3 or -3.

i) If x - 3 = 3, then x = 6.
ii) If x - 3 = -3, then x = 0.

2) If |x - 3| - 2 = -1, then |x - 3| = 1. So x - 3 could be 1 or -1.

i) If x - 3 = 1, then x = 4.
ii) If x - 3 = -1, then x = 2.

We see that the equation has 4 solutions.

Answer:E

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by Max@Math Revolution » Wed Jan 24, 2018 12:21 am
=>

||x-3|-2|=1
=> |x-3|-2=±1
=> |x-3|=2±1
=> |x-3|=1 or |x-3|=3
=> x-3=±1 or x-3=±3
=> x=3±1 or x=3±3
=> x=2, x = 4, x = 0 or x = 6

Thus, the equation has four solutions.

Therefore, the answer is E.

Answer: E
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