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

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Math Revolution GMAT Instructor
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How many solutions has the equation ||x-3|-2|=1?  [#permalink]

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New post 22 Jan 2018, 01:04
2
5
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

61% (01:34) correct 39% (01:30) wrong based on 255 sessions

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[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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How many solutions has the equation ||x-3|-2|=1?  [#permalink]

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New post 22 Jan 2018, 09:55
MathRevolution 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


\(||x-3|-2|=1\)

\(=>|x-3|-2=1\) or \(|x-3|-2=-1\)

\(=>|x-3|=3\) or \(|x-3|=1\)

\(=>x-3=3\) or \(x-3=-3\) or \(x-3=1\) or \(x-3=-1\)

therefore \(x=6\) or \(0\) or \(4\) or \(2\). hence \(4\) solutions

Option E
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Re: How many solutions has the equation ||x-3|-2|=1?  [#permalink]

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New post 23 Jan 2018, 08:58
MathRevolution 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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Re: How many solutions has the equation ||x-3|-2|=1?  [#permalink]

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New post 24 Jan 2018, 01:31
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1
=>

\(||x-3|-2|=1\)
\(⇔|x-3|-2=±1\)
\(⇔|x-3|=2±1\)
\(⇔|x-3|=1 or |x-3|=3\)
\(⇔x-3=±1 or x-3=±3\)
\(⇔x=3±1\) 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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Re: How many solutions has the equation ||x-3|-2|=1?  [#permalink]

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New post 15 Jun 2019, 09:02
MathRevolution 0 and 2 do not satisfy the ranges that are defined for x. Do they still count as solutions? x=2 when 1<x<3 and x=0 when x<1 (and x<3) . These don't satisfy the ranges. Please help. THanks!
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Re: How many solutions has the equation ||x-3|-2|=1?  [#permalink]

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New post 14 Jul 2019, 03:23
GittinGud wrote:
MathRevolution 0 and 2 do not satisfy the ranges that are defined for x. Do they still count as solutions? x=2 when 1<x<3 and x=0 when x<1 (and x<3) . These don't satisfy the ranges. Please help. THanks!


||x-3|-2|=1
    1) X > 3 -------> X - 3 = +ve
      \(|x-3-2|=1\) ----> \(|x-5|=1\)
        \(x-5 = 1\)
          x = 6
        OR
        \(x-5 = -1\)
          x = 4
    Because both the values are greater than 3, both are allowed (4 & 6)

    2) X < 3 -------> X - 3 = -ve
      \(||x-3|-2|=1\) ----> \(|3-x-2|=1\) ----> \(|1-x| = 1\)
        \(1 - x = 1\)
          x = 0
        OR
        \(1 - x = -1\)
          x = 2
    Because both the values are less than 3, both are allowed (0 & 2)

The permissible values of x do satisfy the range.
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Re: How many solutions has the equation ||x-3|-2|=1?   [#permalink] 14 Jul 2019, 03:23
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