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

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4865
GPA: 3.82
How many solutions has the equation ||x-3|-2|=1? [#permalink]

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22 Jan 2018, 00:04
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Difficulty:

55% (hard)

Question Stats:

55% (00:45) correct 45% (00:58) wrong based on 86 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
[Reveal] Spoiler: OA

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

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22 Jan 2018, 08:55
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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23 Jan 2018, 07:58
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.

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

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24 Jan 2018, 00:31
=>

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

Thus, the equation has four solutions.

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Re: How many solutions has the equation ||x-3|-2|=1?   [#permalink] 24 Jan 2018, 00:31
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