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Manager  Joined: 19 Oct 2011
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Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Is x > 0?

(1) |x + 3| = 4x – 3
(2) |x – 3| = |2x – 3|
Math Expert V
Joined: 02 Sep 2009
Posts: 61211
Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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1
50
Is x > 0?

(1) $$|x+3|=4x-3$$ --> LHS is an absolute value, which is always non-negative ($$|some \ expression|\geq{0}$$), so RHS must also be non-negative --> $$4x-3\geq{0}$$ --> $$x\geq{\frac{3}{4}}$$, hence $$x>0$$. Sufficient. You can see here that you don't even need to find exact value(s) of x to answer the question.

(2) $$|x-3|=|2x-3|$$. Square both sides: $$(x-3)^2=(2x-3)^2$$ --> $$(2x-3)^2-(x-3)^2=0$$. Apply $$a^2-b^2=(a-b)(a+b)$$, rather than squaring: --> $$x(3x-6)=0$$ --> $$x=0$$ or $$x=2$$. Not sufficient.

Hope it helps.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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7
Bunnel,
For statement one, I understand that you have not solved for x in your solution. I am still learning the abs value and in equalities. Can you share your thoughts on the following solution (solving for x in both statements)

a. |x+3| = 4x -3
|x+3| = x + 3 for x + 3 >= 0 ---> x > -3.
Solving for x,
x + 3 = 4x -3 ---> 6 = 3x ---> x = 2 ( solution accepted since x > -3)

| x + 3| = -(x +3 ) for x + 3 < 0 --> x < -3.
Solving for x,
-x - 3 = 4x -3
5x = 0 ---> x = 0 ( solution discarded as x is not < -3)

A is sufficient.

b. |x -3| = |2x -3|
|x -3| = x -3 for x-3 >= 0 ---> x >= 3
|x -3| = -(x -3) for x-3 < 0 ---> x < 3

|2x -3 | = 2x -3 for 2x -3 >= 0 ----> x >=3/2
|2x -3 | = -(2x -3) for 2x -3 < 0 ----> x < 3/2

So we have 3 ranges

1) x >=3 ==> x - 3 = 2x -3 ====> x = 0 ; discard
2) 3/2 < x < 3 ===> -(x-3) = 2x -3 ====> -x + 3 = 2x -3 ===> x =2 ( accepted solution )
3) x < 3/2 ===> -(x-3) = -(2x -3) ====> x = 3 (discarded the solution)

Let me know if I have solved the question correctly. I know the process is lengthy.

One more question, if we consider the only acceptable solution in statement b i.e., x =2 (case 3/2 < x < 3), then both statements are sufficient independently and we get the answer as option (D).

Regards
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 61211
Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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3
saxenaashi wrote:
Bunnel,
For statement one, I understand that you have not solved for x in your solution. I am still learning the abs value and in equalities. Can you share your thoughts on the following solution (solving for x in both statements)

a. |x+3| = 4x -3
|x+3| = x + 3 for x + 3 >= 0 ---> x > -3.
Solving for x,
x + 3 = 4x -3 ---> 6 = 3x ---> x = 2 ( solution accepted since x > -3)

| x + 3| = -(x +3 ) for x + 3 < 0 --> x < -3.
Solving for x,
-x - 3 = 4x -3
5x = 0 ---> x = 0 ( solution discarded as x is not < -3)

A is sufficient.

b. |x -3| = |2x -3|
|x -3| = x -3 for x-3 >= 0 ---> x >= 3
|x -3| = -(x -3) for x-3 < 0 ---> x < 3

|2x -3 | = 2x -3 for 2x -3 >= 0 ----> x >=3/2
|2x -3 | = -(2x -3) for 2x -3 < 0 ----> x < 3/2

So we have 3 ranges

1) x >=3 ==> x - 3 = 2x -3 ====> x = 0 ; discard
2) 3/2 < x < 3 ===> -(x-3) = 2x -3 ====> -x + 3 = 2x -3 ===> x =2 ( accepted solution )
3) x < 3/2 ===> -(x-3) = -(2x -3) ====> x = 3 (discarded the solution)

Let me know if I have solved the question correctly. I know the process is lengthy.

One more question, if we consider the only acceptable solution in statement b i.e., x =2 (case 3/2 < x < 3), then both statements are sufficient independently and we get the answer as option (D).

Regards

It seems that you understand this method very well.

Everything is correct except the red parts: -(x-3) = -(2x -3) --> x=0 (not x=3), so it's also a valid solution. Hence for (2) you have two valid solutions x=2 and x=0 (just like in my post above), which makes this statement not sufficient.

Hope it's clear.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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dvinoth86 wrote:
Is x > 0?

(1) |x + 3| = 4x – 3
(2) |x – 3| = |2x – 3|

Liked the question? encourage by giving kudos Remember: When you have || on both sides of eqn you do not need to VERIFY the answer by putting them back in eqn BUT when you have || on only one side you MUST VERIFY the answer by putting them back in eqn

(1) |x + 3| = 4x – 3
a) x + 3 = 4x – 3 => x=2 .. VALID
b) -(x + 3) = 4x – 3 => x=0 .. INVALID

Hence Sufficient

(2) |x – 3| = |2x – 3| => x = 0 or 6 .. INVALID
Hence In-Sufficient
Math Expert V
Joined: 02 Sep 2009
Posts: 61211
Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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WholeLottaLove wrote:
Hi!

Why do we not find the positive and negative values for |x+3|=4x-3? Is it because this isn't a <, >, >= problem?

Thanks!

Bunuel wrote:
Is x > 0?

(1) $$|x+3|=4x-3$$ --> LHS is an absolute value, which is always non-negative ($$|some \ expression|\geq{0}$$), so RHS must also be non-negative --> $$4x-3\geq{0}$$ --> $$x\geq{\frac{3}{4}}$$, hence $$x>0$$. Sufficient. You can see here that you don't even need to find exact value(s) of x to answer the question.

(2) $$|x-3|=|2x-3|$$. Square both sides: $$(x-3)^2=(2x-3)^2$$ --> $$(2x-3)^2-(x-3)^2=0$$. Apply $$a^2-b^2=(a-b)(a+b)$$, rather than squaring: --> $$x(3x-6)=0$$ --> $$x=0$$ or $$x=2$$. Not sufficient.

Hope it helps.

One can do this way too, but the way shown in my post is faster.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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2
vikram4689 wrote:
dvinoth86 wrote:
Is x > 0?

(1) |x + 3| = 4x – 3
(2) |x – 3| = |2x – 3|

Liked the question? encourage by giving kudos Remember: When you have || on both sides of eqn you do not need to VERIFY the answer by putting them back in eqn BUT when you have || on only one side you MUST VERIFY the answer by putting them back in eqn

(1) |x + 3| = 4x – 3
a) x + 3 = 4x – 3 => x=2 .. VALID
b) -(x + 3) = 4x – 3 => x=0 .. INVALID

Hence Sufficient

(2) |x – 3| = |2x – 3| => x = 0 or 6 .. INVALID
Hence In-Sufficient

Your statement 1 has two sceneries. Then how does it become sufficient?

The solution quoted by you above has some tacit steps. I'll list them here:

St. 1 says: |x + 3| = 4x – 3 . . . (1)

Case 1: x + 3 > = 0

That is, x > = -3

In this case, |x+3| = (x+3)

So, Equation 1 becomes:

x + 3 = 4x - 3
=> 6 = 3x
=> x = 2

Does this value of x satisfy the condition of Case 1, that x > = -3?

Yes, it does. So, x = 2 is a valid value of x.

Let's now consider

Case 2: x + 3 < 0

That is, x < -3

In this case, |x+3| = -(x+3)

So, Equation 1 becomes:

-(x + 3) = 4x - 3
=> -x - 3 = 4x - 3
=> 0 = 5x
=> x = 0

Does this value of x satisfy the condition of Case 2, that x < -3?

No, it doesn't. So, x = 0 is an INVALID value of x.

Thus, we got only 1 valid value of X from Statement 1 (x = 2) and so, Statement 1 is sufficient to say that x > 0.

Hope this answered your doubt! Japinder
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Quote:
Hi

I am not clear why are we taking ONLY x+3 as x+3>=0 and not x+3<=0?

Whenever you translate an absolute value to 'go both ways' like in that explanation, you need to take an additional step and double check to make sure you haven't come up with an invalid solution. You're correct that you need to consider both x + 3 > 0 and x + 3 < 0. But once you've figured out what that implies about the value of x, double check by plugging x back in. For instance, if x + 3 < 0, we ended up figuring out that x has to equal 0. But that's impossible, because 0 + 3 isn't less than 0. So, we didn't get a valid solution at all, and only the x + 3 > 0 solution worked.
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Math Expert V
Joined: 02 Sep 2009
Posts: 61211
Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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saxenaashi wrote:
Works. Thanks for pointing out the mistake and quick response too. I am just learning this so working ground up, hence I am not hesitant in taking long route initially. After next few questions, I would be tuned to extract the ranges I guess. Is there a condition on squaring the sides of the equation or inequation.

Regards

A. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
$$2<4$$ --> we can square both sides and write: $$2^2<4^2$$;
$$0\leq{x}<{y}$$ --> we can square both sides and write: $$x^2<y^2$$;

But if either of side is negative then raising to even power doesn't always work.
For example: $$1>-2$$ if we square we'll get $$1>4$$ which is not right. So if given that $$x>y$$ then we can not square both sides and write $$x^2>y^2$$ if we are not certain that both $$x$$ and $$y$$ are non-negative.

B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
$$-2<-1$$ --> we can raise both sides to third power and write: $$-2^3=-8<-1=-1^3$$ or $$-5<1$$ --> $$-5^2=-125<1=1^3$$;
$$x<y$$ --> we can raise both sides to third power and write: $$x^3<y^3$$.

So in statement (2) since both parts of expression are non-negative we can safely apply squaring.

Hope it helps.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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terance wrote:
Well, I have a little bit another solution for this problem

Consider (1): if x>-3 then x+3=4x-3, so x=2, sufficient
if x<-3, then -x-3=4x-3, which Leeds to answer x=0, which is wrong because we consider only x<-3
Sum up (1) is sufficient

The saim logic can be applied to the (2), and it is possible to derive that (2) is sufficient

So the answer is D either (1) is sufficient or (2) is sufficient

In your logic, you gave a mistake, because you've missed the root. Posted from GMAT ToolKit

Welcome to GMAT Club.

Unfortunately your answer is not correct: OA for this question is A, not D (you can see it under the spoiler in the initial post).

(2) |x – 3| = |2x – 3| has two roots x=0 and x=2 (just substitute these values to see that they both satisfy the given equation), so you can not get the single numerical value of x, which makes this statement insufficient.

You can refer to above solutions for two different approaches of how to get these roots for (2). Please ask if anything remains unclear.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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nikhil007 wrote:
Hi Bunuel
This is how I solved the 2 statements,
http://campl.us/fWdgqiT5EsK

and I get 0 & 2 for both statements, can u tell me what wrong am I doing?

Posted from my mobile device

If you substitute x=0 in |x + 3| = 4x – 3 you'll get: LHS=|x + 3|=3 and RHS=4x – 3=-3, thus $$LHS\neq{RHS}$$, which means that x=0 is not the root of the given equation.

When expanding |x+3|:

When x<-3, then |x+3|=-(x+3), so in this case we'll have -(x+3)=4x-3 --> x=0 --> discard this value since 0 is not less than -3 (we consider the range when x<-3).

When x>=-3, then |x+3|=x+3, so in this case we'll have x+3=4x-3 --> x=2 --> this value of x is OK since 2>-3.

So, |x + 3| = 4x - 3 has only one root, x=2.

Hope it's clear.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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WholeLottaLove wrote:
Bunuel wrote:
$$|x+3|=4x-3$$. Check point is at $$x=-3$$ (check point, is the value of x for which the value of an expression in modulus equals to zero).

When $$x\leq{-3}$$, then $$x+3<0$$, thus $$|x+3|=-(x+3)$$. So, in this case we have $$-(x+3)=4x-3$$ --> $$x=0$$ --> discard this solution since we consider the range when $$x\leq{-3}$$.

When $$x>{-3}$$, then $$x+3>0$$, thus $$|x+3|=x+3$$. So, in this case we have $$x+3=4x-3$$ --> $$x=2$$ --> since $$x=2>-3$$, then this solution is valid.

So, we have that $$|x+3|=4x-3$$ has only one root: $$x=2$$.

Hope it's clear.

Why do we discard that solution?

We consider the range $$x\leq{-3}$$. x=0 is out of this range.

Check absolute values chapter of math book: math-absolute-value-modulus-86462.html

Hope it helps.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Mo2men wrote:
Thanks GMATGuruNY.

In this type of questions asking for sign of x, does this mean that I can deal with such statement in same manner like statement 1? or do I need open modulus to be sure from the sign?

Since the question stem asks whether x>0 -- and an equation with absolute value on both sides can have both a positive and a nonpositive solution -- I recommend that you solve the equation, either by opening up the modulus or by squaring both sides.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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topper97 wrote:
Bunuel does it mean that in all questions based on absolute values, one solution of it would be 0?

I'm clear with Stat1 but in stat 2, I used the formula by squaring both sides of absolute figures to find the inequality range. Here is what I did.

|x−3|=|2x−3|
Square both sides: (x−3)^2=(2x−3)^2
x^2-6x+9=4x^2-12x+9
3x^2=6x
Therefore, x=2. So I chose option D. Can you help me to find what is the error? I'm not getting it. All I did was use the formula.

1. Of course 0 is NOT a solution of all absolute value questions. Why would it be?

2. You solved 3x^2 = 6x incorrectly. You cannot reduce 3x^2 = 6x by x because x can be 0 and we cannot divide by 0. By doing so you loose a root, namely x = 0.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Bunuel wrote:
saxenaashi wrote:
Bunnel,
For statement one, I understand that you have not solved for x in your solution. I am still learning the abs value and in equalities. Can you share your thoughts on the following solution (solving for x in both statements)

a. |x+3| = 4x -3
|x+3| = x + 3 for x + 3 >= 0 ---> x > -3.
Solving for x,
x + 3 = 4x -3 ---> 6 = 3x ---> x = 2 ( solution accepted since x > -3)

| x + 3| = -(x +3 ) for x + 3 < 0 --> x < -3.
Solving for x,
-x - 3 = 4x -3
5x = 0 ---> x = 0 ( solution discarded as x is not < -3)

A is sufficient.

b. |x -3| = |2x -3|
|x -3| = x -3 for x-3 >= 0 ---> x >= 3
|x -3| = -(x -3) for x-3 < 0 ---> x < 3

|2x -3 | = 2x -3 for 2x -3 >= 0 ----> x >=3/2
|2x -3 | = -(2x -3) for 2x -3 < 0 ----> x < 3/2

So we have 3 ranges

1) x >=3 ==> x - 3 = 2x -3 ====> x = 0 ; discard
2) 3/2 < x < 3 ===> -(x-3) = 2x -3 ====> -x + 3 = 2x -3 ===> x =2 ( accepted solution )
3) x < 3/2 ===> -(x-3) = -(2x -3) ====> x = 3 (discarded the solution)

Let me know if I have solved the question correctly. I know the process is lengthy.

One more question, if we consider the only acceptable solution in statement b i.e., x =2 (case 3/2 < x < 3), then both statements are sufficient independently and we get the answer as option (D).

Regards

It seems that you understand this method very well.

Everything is correct except the red parts: -(x-3) = -(2x -3) --> x=0 (not x=3), so it's also a valid solution. Hence for (2) you have two valid solutions x=2 and x=0 (just like in my post above), which makes this statement not sufficient.

Hope it's clear.

Works. Thanks for pointing out the mistake and quick response too. I am just learning this so working ground up, hence I am not hesitant in taking long route initially. After next few questions, I would be tuned to extract the ranges I guess. Is there a condition on squaring the sides of the equation or inequation.

Regards
Intern  Joined: 26 Jan 2012
Posts: 1
Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Well, I have a little bit another solution for this problem

Consider (1): if x>-3 then x+3=4x-3, so x=2, sufficient
if x<-3, then -x-3=4x-3, which Leeds to answer x=0, which is wrong because we consider only x<-3
Sum up (1) is sufficient

The saim logic can be applied to the (2), and it is possible to derive that (2) is sufficient

So the answer is D either (1) is sufficient or (2) is sufficient

In your logic, you gave a mistake, because you've missed the root. Posted from GMAT ToolKit
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Hi Bunuel
This is how I solved the 2 statements,
http://campl.us/fWdgqiT5EsK

and I get 0 & 2 for both statements, can u tell me what wrong am I doing?

Posted from my mobile device
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Hi!

Why do we not find the positive and negative values for |x+3|=4x-3? Is it because this isn't a <, >, >= problem?

Thanks!

Bunuel wrote:
Is x > 0?

(1) $$|x+3|=4x-3$$ --> LHS is an absolute value, which is always non-negative ($$|some \ expression|\geq{0}$$), so RHS must also be non-negative --> $$4x-3\geq{0}$$ --> $$x\geq{\frac{3}{4}}$$, hence $$x>0$$. Sufficient. You can see here that you don't even need to find exact value(s) of x to answer the question.

(2) $$|x-3|=|2x-3|$$. Square both sides: $$(x-3)^2=(2x-3)^2$$ --> $$(2x-3)^2-(x-3)^2=0$$. Apply $$a^2-b^2=(a-b)(a+b)$$, rather than squaring: --> $$x(3x-6)=0$$ --> $$x=0$$ or $$x=2$$. Not sufficient.

Hope it helps.
Senior Manager  Joined: 13 May 2013
Posts: 396
Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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Hmmm...

When I solve for the pos. and neg. values of |x+3|=4x-3 I get:

I. x+3=4x-3 ==> -3x=-6 ==> x=2
II. x+3=-(4x-3) ==> x+3=-4x+3 ==> 5x=0 ==> x=0

So here in my presumably incorrect simplification, I have x=2 and x=0 in which case we can't be sure if x>0

For many abs. value questions it seems that you have to find the positive and negative cases each equation. I get that in this case, |x+3|=4x-3 means that 4x-3 is positive but why for other, similar questions, is solving for the positive and negative cases necessary?

Bunuel wrote:
WholeLottaLove wrote:
Hi!

Why do we not find the positive and negative values for |x+3|=4x-3? Is it because this isn't a <, >, >= problem?

Thanks!

Bunuel wrote:
Is x > 0?

(1) $$|x+3|=4x-3$$ --> LHS is an absolute value, which is always non-negative ($$|some \ expression|\geq{0}$$), so RHS must also be non-negative --> $$4x-3\geq{0}$$ --> $$x\geq{\frac{3}{4}}$$, hence $$x>0$$. Sufficient. You can see here that you don't even need to find exact value(s) of x to answer the question.

(2) $$|x-3|=|2x-3|$$. Square both sides: $$(x-3)^2=(2x-3)^2$$ --> $$(2x-3)^2-(x-3)^2=0$$. Apply $$a^2-b^2=(a-b)(a+b)$$, rather than squaring: --> $$x(3x-6)=0$$ --> $$x=0$$ or $$x=2$$. Not sufficient.

Hope it helps.

One can do this way too, but the way shown in my post is faster.
Math Expert V
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Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|  [#permalink]

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$$|x+3|=4x-3$$. Check point is at $$x=-3$$ (check point, is the value of x for which the value of an expression in modulus equals to zero).

When $$x\leq{-3}$$, then $$x+3<0$$, thus $$|x+3|=-(x+3)$$. So, in this case we have $$-(x+3)=4x-3$$ --> $$x=0$$ --> discard this solution since we consider the range when $$x\leq{-3}$$.

When $$x>{-3}$$, then $$x+3>0$$, thus $$|x+3|=x+3$$. So, in this case we have $$x+3=4x-3$$ --> $$x=2$$ --> since $$x=2>-3$$, then this solution is valid.

So, we have that $$|x+3|=4x-3$$ has only one root: $$x=2$$.

Hope it's clear.
_________________ Re: Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|   [#permalink] 04 Jun 2013, 07:42

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