If |x| = 3x - 2, then x = ?

Question

A. 1
B. 1/2
C. 1 and 1/2
D. -1/2
E. -1

Soln
 When we proceed to solve this we get two eq:
 x = 3x - 2 .... (i) when x > 0
-x = 3x - 2 .... (ii) when x < 0

Therefore we get two results x = 1 and x = 1/2 from (i) and (ii) resp.
Lets plug the values back into the eq.

|1| = 3*1 - 2 
==> 1 = 1
so (i) holds good.

|1/2| = 3*(1/2) - 2
==> 1/2 = -1/2 this is incorrect, hence soln from eq (ii) does not hold good.

The soln to the eq |x| = 3x - 2 is 1 only.

Quant guru's let me know if there is a flaw in my calculation method.

Thanks!

bagrettin · Accepted Answer

There is no big flaws, but it is not fully mathematically correct. When opening the absolute value you should add constraints which you use. In other words, \(|x|=x\) if \(x>=0\)

So, the precise solution is as follows:

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

\(x=3x-2 , x>=0\)
\(-x=3x-2, x<0\)

\(2x=2, x>=0\)
\(4x=2, x<0\)

\(x=1, x>=0\)
\(x=0.5, x<0\)

So, the only answer is x=1.
Good luck! Try to solve some more problems with absolute values (both equations and inequalities)

KarishmaB · Answer

I know it's fun to solve when you know what you are doing (and more importantly, why you are doing it) and the graph method is especially satisfying but don't forget a very basic trick in such questions. When you have values for x in the options, and an equation in the question, we could just plug the values in. Since x = 1 satisfies the equation and x = 1/2 doesn't, answer has to be (A).

Bunuel · Answer

|x| = 3x – 2;

The left hand side is an absolute value, which is always non negative, hence the right hand side must also be non-negative:  3x-2\geq{0} ;

x\geq{\frac{2}{3}} ;

x  positive, so  |x|=x ;

x=3x-2 ;

x=1 .

Answer: A.

pike · Answer

I think one other point is that visually we should know this has only one solution.

Visualize y = |x|

https://i.imgur.com/DcSMn.gif

Visualize y = 3x - 2

https://i.imgur.com/FYKtF.gif

These will only intersect once!

https://i.imgur.com/0lmst.gif

On the other hand, consider:

|2x – 3| – 4 = 3
|2x – 3| = 7

We can see straight away this has two solutions. Visualize the left hand side and the right hand side:

https://i.imgur.com/qyVg2.gif

mave23 · Answer

Wow thanks, you are correct i had overlooked this option.. do you think such questions will come in the real gmat, where plugging the value may reveal the answer?

Thanks.

KarishmaB · Answer

Sure it is possible. That is why it is good to keep this in mind since a relatively tricky question can be easily solved this way. Though, GMAC is wise to such tricks and since these questions don't involve much work on your part, don't expect the options to help you often (nevertheless, you should always keep one eye on the options), at least not right in the beginning. If you do get such a question, it may not be a very high level question.

igotthis · Answer

Absolute values are easy. The hard part is remembering to substitute back into the original question to check if it satisfies the equation.

Here are some rules:
|x|>=0
|x| = sqrt(x)^2
|0| = 0
|-x|=|x|
|x|+|y|>= |x+y|

Absolute values can also be thought of as the distance between 2 points.

Remember to always always always check whether your solution satisfies the equation!!

email2vm · Answer

Here is the sol. I have attached a screenshot. HTH

EMPOWERgmatRichC · Answer

Hi All,

This question can be solved by TESTing THE ANSWERS.

Let's start with the easiest of the possibilities: 1. Does X=1 fit the given equation?

|1| = 3(1) - 2..... 1 = 1
This is clearly a possible solution, so it must be in the correct answer. Eliminate Answers B, D and E.

The only option that remains is X = 1/2. Does X = 1/2 fit the given equation?

|1/2| = 3(1/2) - 2..... 1/2 = -1/2.
This is NOT mathematically correct, so X = 1/2 is NOT a potential solution. Eliminate Answer C.

Final Answer:  A

GMAT assassins aren't born, they're made,
Rich

MBAHOUSE · Answer

If |x| = 3x - 2, then x = ?

A. 1
B. 1/2
C. 1 and 1/2
D. -1/2
E. -1

ThatDudeKnows · Answer

This question highlights two really big test-taking techniques...hint: neither has anything to do with absolute values!!

First, the GC timer shows that FORTY-SIX percent of people are missing this question...YIKES!! If you picked C, why didn't you take the little bit of extra time to check to make sure that 1/2 works? You could have proven/disproven it in a matter of seconds and saved yourself from missing a question that you should have gotten right.

Second, Plugging In The Answers (PITA) takes 20 seconds and is soooo easy; why would we mess around with solving for x?
There are only four possible values listed among the answer choices. Pick one and test it. 1 looks the easiest.
|1| = 3(1) - 2
1 = 3-2
1=1 
Yep. B, D, and E are out.
Let's test 1/2.
|0.5| = 3(0.5) - 2
0.5 = 1.5 - 2
0.5 = -0.5
Nope. C is out.

Answer choice A.

ThatDudeKnowsPITA

ScottTargetTestPrep · Answer

We can solve the equations x = 3x - 2 and x = -(3x - 2) and check for extraneous solutions, but I think testing answer choices is a better approach for this question. We can immediately eliminate answer choices D and E because both of them make the right hand side of the equation negative, and an absolute value can never be negative. For the remaining answer choices, we just need to test x = 1 and x = 1/2.

Let's substitute x = 1 in the given equation:

\Rightarrow  |x| = 3x - 2

\Rightarrow  Is |1| = 3(1) - 2 ?

\Rightarrow  Is 1 = 3 - 2 ? Yes.

So, x = 1 is a solution to the given equation.

Next, let's substitute x = 1/2 in the given equation:

\Rightarrow  |x| = 3x - 2

\Rightarrow  Is |1/2| = 3(1/2) - 2 ?

\Rightarrow  Is 1/2 = 3/2 - 2 ?

\Rightarrow  Is 1/2 = -1/2 ? No.

We see that of the given numbers, only x = 1 is a solution to the given equation.

Answer: A

BrushMyQuant · Answer

|x| = 3x - 2, then x = ?

Let's solve the problem using two methods:

Method 1: Substitution

Let's take each answer choice and substitute in the question and check which one satisfies

A. 1
|1| = 3*1 - 2 = 3 - 2 = 1
=> 1 = 1
=> TRUE

We don't need to solve further as 1 is an option choice, but solving to complete the solution

B. 1/2
|1/2| = 3*1/2 - 2 = 1.5 - 2 = -0.5
=> 0.5 = -0.5
=> FALSE

C. 1 and 1/2
=> FALSE as x cannot be 1/2

D. -1/2
|-1/2| = 3*-1/2 - 2 = -1.5 - 2 = -3.5
=> 0.5 = -3.5
=> FALSE

E. -1
|-1| = 3*-1 - 2 = -3 - 2 = -5
=> 1 = -5
=> FALSE

So, Answer will be A

Method 2: Algebra

As we have |x| in the equation so we will have two cases

As we have \|x\| in the equation so we will have two cases
-Case 1: x ≥ 0 => \|x\| = x => x = 3x - 2 => 2x = 2 => x = 1 And 1 ≥ 0 => This is a Solution We don't need to solve further as 1 is an option choice, but solving to complete the solution	-Case 2: x < 0 => \|x\| = -x => -x = 3x - 2 => 4x = 2 => x = \(\frac{2}{4}\) = \(\frac{1}{2}\) But condition was x < 0 => x = \(\frac{1}{2}\) is NOT a SOLUTION

So, Answer will be A
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

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If |x| = 3x - 2, then x = ?

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