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# How many real numbers x satisfy ||x-3|-4|=4?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7367
GMAT 1: 760 Q51 V42
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How many real numbers x satisfy ||x-3|-4|=4?  [#permalink]

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04 Dec 2017, 01:36
1
8
00:00

Difficulty:

45% (medium)

Question Stats:

60% (01:27) correct 40% (01:40) wrong based on 246 sessions

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[GMAT math practice question]

How many real numbers $$x$$ satisfy $$||x-3|-4|=4$$?

A. 1
B. 2
C. 3
D. 4
E. 5

_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $149 for 3 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: 1214 Location: India GPA: 3.82 How many real numbers x satisfy ||x-3|-4|=4? [#permalink] ### Show Tags 04 Dec 2017, 05:35 3 MathRevolution wrote: [GMAT math practice question] How many real numbers $$x$$ satisfy $$||x-3|-4|=4$$? A. 1 B. 2 C. 3 D. 4 E. 5 $$||x-3|-4|=4$$ $$=>|x-3|-4=4$$ or $$|x-3|-4=-4$$ $$=>|x-3|=8$$ or $$|x-3|=0$$ $$=>x-3=8$$ or $$x-3=-8$$ or $$x=3$$ $$=>x=11$$ or $$x=-5$$ or $$x=3$$ Hence $$3$$ real values satisfy the equation Option C Math Expert Joined: 02 Aug 2009 Posts: 7684 Re: How many real numbers x satisfy ||x-3|-4|=4? [#permalink] ### Show Tags 04 Dec 2017, 06:14 1 1 MathRevolution wrote: [GMAT math practice question] How many real numbers $$x$$ satisfy $$||x-3|-4|=4$$? A. 1 B. 2 C. 3 D. 4 E. 5 if you have POSITIVE values on both sides, you can square both sides.. $$||x-3|-4|=4............(|x-3|-4)^2=4^2.........(x-3)^2+4^2-2*4*|x-3|=4^2.........(x-3)^2=8|x-3|$$ you can square both sides again $$(x-3)^2=8|x-3|.............(x-3)^4=8^2*(x-3)^2..............(x-3)^4-8^2*(x-3)^2=0.................(x-3)^2((x-3)^2-8^2)=0.....................(x-3)^2(x-3-8)(x-3+8)=0............(x-3)^2(x-11)(x+5)=0$$ so roots of equation or the different values of x are 3,11,-5 ans 3 different values C _________________ Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7367 GMAT 1: 760 Q51 V42 GPA: 3.82 How many real numbers x satisfy ||x-3|-4|=4? [#permalink] ### Show Tags 06 Dec 2017, 01:09 2 => $$| |x-3| - 4 | = 4$$ $$⇔ |x-3| - 4= ± 4$$ $$⇔ |x-3| = 4 ± 4$$ $$⇔ |x-3| = 8$$ or $$|x-3| = 0$$ $$⇔ x-3 = ±8$$ or $$x-3 = 0$$ $$⇔ x = 3 ± 8$$ or $$x = 3$$ $$⇔ x = 11, x = -5$$ or $$x = 3$$ The equation has 3 solutions. Therefore, the answer is C. Answer : C _________________ 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$149 for 3 month Online Course"
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How many real numbers x satisfy ||x-3|-4|=4?  [#permalink]

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27 Mar 2018, 04:58
1
MathRevolution wrote:
[GMAT math practice question]

How many real numbers $$x$$ satisfy $$||x-3|-4|=4$$?

A. 1
B. 2
C. 3
D. 4
E. 5

Open the outer Modulus and make the RHS + or -

|x-3|-4=+or-4. Now isolate the remaining Mod.

|x-3|=4+-4. So |x-3|=8. Or |x-3|=0. Both 0 and 8 are non negative, so no need to discard either formula. Now use both formulas to retrieve solutions.

(1) |x-3|=8 x=-5 or 11

(2) |x-3|=0 x=3

So 3 solutions. Good question.
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How many real numbers x satisfy ||x-3|-4|=4?   [#permalink] 27 Mar 2018, 04:58
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