Is x > 0? (1) |x + 3| = 4x – 3 (2) |x – 3| = |2x – 3|

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

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.

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.

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

Hi Bunuel
This is how I solved the 2 statements,
https://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.

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!

One can do this way too, but the way shown in my post is faster.

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?

\(|x+3|=4x-3\). 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.

Hi Bunnel,
I've been struggling a lot in inequalities.. I've a question pertaining to solution provided by saxenaashi ...


1) x >=3 ==> x - 3 = 2x -3 ====> x = 0 ; discard[Understood this part]2) 3/2 < x < 3 ===> -(x-3) = 2x -3 ====> -x + 3 = 2x -3 ===> x =2 ( accepted solution )[Did not understand, Why -ve sign is here as the x range is positive]3) x < 3/2 ===> -(x-3) = -(2x -3) ====> x = 3 (discarded the solution)[Understood]
Another thing is 3/2(inclusive) is not at all considered in the range in any of the three checkpoints.

What am i missing here and how to solve these questions effectively

Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\).

SOLUTION:

We have two transition points for \(|x-3|=|2x-3|\): \(x=\frac{3}{2}\) and \(x=3\). Thus three ranges to check:
1. \(x<\frac{3}{2}\);
2. \(\frac{3}{2}\leq{x}\leq{3}\);
3. \(3<x\)

Note that it does not matter in which range(s) you include the transition points with "=" sign as long you include them.

1. When \(x<\frac{3}{2}\), then \(x-3\) is negative and \(2x-3\) is negative too, thus \(|x-3|=-(x-3)\) and \(|2x-3|=-(2x-3)\).

Therefore for this range \(|x-3|=|2x-3|\) transforms to \(-(x-3)=-(2x-3)\) --> \(x=0\). This solution is OK, since \(x=0\) is in the range we consider (\(x<\frac{3}{2}\)).

2. When \(\frac{3}{2}\leq{x}\leq{3}\), then \(x-3\) is negative and \(2x-3\) is positive, thus \(|x-3|=-(x-3)\) and \(|2x-3|=2x-3\).

Therefore for this range \(|x-3|=|2x-3|\) transforms to \(-(x-3)=2x-3\) --> \(x=2\). This solution is OK, since \(x=2\) is in the range we consider (\(\frac{3}{2}\leq{x}\leq{3}\)).

3. When \(3<x\), then \(x-3\) is positive and \(2x-3\) is positive too, thus \(|x-3|=x-3\) and \(|2x-3|=2x-3\).

Therefore for this range \(|x-3|=|2x-3|\) transforms to \(x-3=2x-3\) --> \(x=0\). This solution is NOT OK, since \(x=0\) is NOT in the range we consider (\(3<x\)).

Thus \(|x-3|=|2x-3|\) has two solutions \(x=0\) and \(x=2\).

Hope it's clear.

P.S. Though for this particular question I still suggest another approach shown in my post here: is-x-0-1-x-3-4x-3-2-x-3-2x-127978.html#p1048512

Is x > 0?

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

(1) |x + 3| = 4x – 3

Well, we know that (4x-3) must be greater than or equal to zero: 4x-3 >= 0 ===> 4x>=3 ===> x>=3/4

So, for #1 I would say that it is sufficient. For 4x-3 to be positive x must be greater than or equal to 3/4.
SUFFICIENT

(2) |x – 3| = |2x – 3|

We don't know if both sides are positive so we can't take the square root. So: |x – 3| = |2x – 3|

(x-3)=(2x-3) ===> x-3=2x-3 ===> 3x=0 ===> x=0
OR
(x-3)=-2x+3 ===> x-3=-2x+3 ===> 3x=6 ===> x=2

Here, we have two valid solutions.
INSUFFICIENT

EDIT: Apparently you CAN square both sides in #2. I thought you had to know that, for example (x-3) and (2x-3) are positive in order for you to be able to square both sides?