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

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Math Revolution GMAT Instructor
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How many real numbers x satisfy ||x-3|-4|=4? [#permalink]

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04 Dec 2017, 00:36
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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
[Reveal] Spoiler: OA

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

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04 Dec 2017, 04:35
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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

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Re: How many real numbers x satisfy ||x-3|-4|=4? [#permalink]

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04 Dec 2017, 05:14
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
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Math Revolution GMAT Instructor
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How many real numbers x satisfy ||x-3|-4|=4? [#permalink]

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06 Dec 2017, 00:09
=>

$$| |x-3| - 4 | = 4$$
$$⇔ |x-3| - 4= ± 4$$
$$⇔ |x-3| = 4 ± 4$$
$$⇔ |x-3| = 8$$ 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.

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How many real numbers x satisfy ||x-3|-4|=4?   [#permalink] 06 Dec 2017, 00:09
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