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

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22 Jan 2018, 01:04
2
5
00:00

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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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Retired Moderator Joined: 25 Feb 2013 Posts: 1178 Location: India GPA: 3.82 How many solutions has the equation ||x-3|-2|=1? [#permalink] ### Show Tags 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 Target Test Prep Representative Status: Head GMAT Instructor Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2815 Re: How many solutions has the equation ||x-3|-2|=1? [#permalink] ### Show Tags 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 _________________ # Jeffrey Miller Head of GMAT Instruction Jeff@TargetTestPrep.com 122 Reviews 5-star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8017 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: How many solutions has the equation ||x-3|-2|=1? [#permalink] ### Show Tags 24 Jan 2018, 01:31 1 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: How many solutions has the equation ||x-3|-2|=1?  [#permalink]

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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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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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