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Manager  Joined: 07 Nov 2009
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Is sqrt ((x-3)^2) = 3-x?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 58% (01:29) correct 42% (01:55) wrong based on 288 sessions

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Is sqrt ((x-3)^2) = 3-x?

(1) x not equal to 3
(2) -x|x| > 0

Let me rephrase it a bit to test my basics:
is sqrt ((x-3)^2) = x-3?

--> |x-3| = x-3
This will be true only if x>= 3 .. Am i correct? Math Expert V
Joined: 02 Sep 2009
Posts: 53734
Re: Redesign sqrt ((x-3)^2) = 3-x  [#permalink]

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rohitgoel15 wrote:
This question has been discussed many times ...

is sqrt ((x-3)^2) = 3-x?
1) x not equal to 3
2) -x|x| > 0

Let me rephrase it a bit to test my basics:
is sqrt ((x-3)^2) = x-3? Incorrect

--> |x-3| = x-3
This will be true only if x>= 3 .. Am i correct? Correct

Remember: $$\sqrt{x^2}=|x|$$.

$$\sqrt{(x-3)^2}=|x-3|$$. So the question becomes is $$|x-3|=3-x$$.

When $$x>3$$, then RHS (right hand side) is negative, but LHS (absolute value) is never negative, hence in this case equations doesn't hold true.

When $$x\leq{3}$$, then $$LHS=|x-3|=-x+3=3-x=RHS$$, hence in this case equation holds true.

Basically question asks is $$x\leq{3}$$?

(1) x is not equal to 3. Clearly insufficient.

(2) $$-x|x| >0$$, basically this inequality implies that $$x<0$$, hence $$x<3$$. Sufficient.

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Manager  Joined: 13 Dec 2009
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Re: Redesign sqrt ((x-3)^2) = 3-x  [#permalink]

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rohitgoel15 wrote:
This question has been discussed many times ...

is sqrt ((x-3)^2) = 3-x?
1) x not equal to 3
2) -x|x| > 0

Let me rephrase it a bit to test my basics:
is sqrt ((x-3)^2) = x-3?

--> |x-3| = x-3
This will be true only if x>= 3 .. Am i correct? |x-3| = 3-x => we have to prove if x <= 3

Stmt1: x not equal to 3 doesnt say if x <3 or >3 so insufficient.
Stmt2: -x |x| > 0
In this case |x| is always going to be postive and to make the expression positive it means -x is also positive so -x>0
or x < 0 from this we can say dat is x <0 it must be < 3 too hence answer is B
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pl help
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Re: gmat prep_inequality  [#permalink]

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2
1
One important thing to know about the square root of a square term:
(1) If it's a number under the square root sign, the answer is only the positive root: sqrt[(-3)^2] = sqrt[(3)^2] = sqrt = 3.
(2) If there are variables under the square root sign, you must consider the fact that the squared "thing" may have been either pos or neg to begin with: sqrt[(-x)^2] = sqrt[(x)^2] = |x|. We are still looking for the positive root, but we don't know whether +x or -x is actually greater than zero--that depends on the sign of x.

So for this question: Is sqrt[(x-3)^2] = 3-x?

Rephrase to: Is |x-3| = 3-x?

If stuff in the absolute value sign is positive or zero, this becomes:
Is x-3 = 3-x?
Is 2x = 6?
Is x = 3?

If stuff in the absolute value sign is negative, this becomes:
Is -(x-3) = 3-x?
Is -x+3 = 3-x?
The answer is yes for all x, but remember this was just "all x" such that x-3 was negative, or x < 3: So we ask "Is x < 3?"

Put the two cases together for the final rephrase: "Is x =<3 ? "

(1) x is not 3, but no info on whether it is less than or greater than 3. INSUFF.
(2) -x|x| > 0.
|x| is positive, so -x would have to be positive too (pos*pos > 0, but neg*pos < 0). Thus, x is negative. If x is negative, it is definitely less than 3. The answer is definitely Yes. SUFF.

The answer is B.
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Re: gmat prep_inequality  [#permalink]

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Thanks esledge . I have got one important take away.
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Is √(x-3)^2 = 3 - x ?  [#permalink]

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Is $$\sqrt{(x-3)^2}$$ = 3 - x

1. x $$\neq$$ 3
2. -x |x|>0

Originally posted by naeln on 31 Jul 2015, 09:24.
Last edited by ENGRTOMBA2018 on 31 Jul 2015, 09:36, edited 1 time in total.
Formatted the question
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Re: Is √(x-3)^2 = 3 - x ?  [#permalink]

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naeln wrote:
Is $$\sqrt{(x-3)^2}$$ = 3 - x

1. x doesn't equal 3
2. -x |x|>0

$$\sqrt{(x-3)^2}$$ = 3 - x ONLY IF, $$x\leq3$$

FYI, $$\sqrt{a^2} = |a|$$

Per statement 1, $$x\neq3$$ but is x<3 ? No information. Thus this statement is not sufficient.

Per statement 2, -x|x| > 0 ---> only case possible is for x<0. Thus we have a definite yes for $$x \leq 3$$. B is the correct answer.
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Re: Is sqrt ((x-3)^2) = 3-x?  [#permalink]

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_________________ Re: Is sqrt ((x-3)^2) = 3-x?   [#permalink] 01 Oct 2018, 06:06
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