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# If zy < xy < 0, is |x - z | + |x| = |z|?

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If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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17 Sep 2010, 12:10
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If zy < xy < 0, is |x - z | + |x| = |z|?

(1) z < x
(2) y > 0
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Joined: 02 Sep 2009
Posts: 47033
Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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17 Sep 2010, 15:58
11
23
This is not a good question, (well at least strange enough) as neither of statement is needed to answer the question, stem is enough to do so. This is the only question from the official source where the statements aren't needed to answer the question. I doubt that such a question will occur on real test but if it ever happens then the answer would be D.

If $$zy<xy<0$$ is $$|x-z|+|x| = |z|$$

Look at the inequality $$zy<xy<0$$:

We can have two cases:

A. If $$y<0$$ --> when reducing we should flip signs and we'll get: $$z>x>0$$.
In this case: as $$z>x$$ --> $$|x-z|=-x+z$$; as $$x>0$$ and $$z>0$$ --> $$|x|=x$$ and $$|z|=z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$-x+z+x=z$$ --> $$0=0$$, which is true.

And:

B. If $$y>0$$ --> when reducing we'll get: $$z<x<0$$.
In this case: as $$z<x$$ --> $$|x-z|=x-z$$; as $$x<0$$ and $$z<0$$ --> $$|x|=-x$$ and $$|z|=-z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$x-z-x=-z$$ --> $$0=0$$, which is true.

So knowing that $$zy<xy<0$$ is true, we can conclude that $$|x-z|+|x| = |z|$$ will also be true. Answer should be D even not considering the statements themselves.

As for the statements:

Statement (1) says that $$z<x$$, hence we have case B.

Statement (2) says that $$y>0$$, again we have case B.

Hope it helps.
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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17 Sep 2010, 12:33
5
Hi, rgtiwari
Draw a number line and put 0 in the middle of it. now look at the data you are given. zy<xy<0.
we don't know if y is positive or negative. if y>0 than z is on the left of x which is on the left of zero => z<x and both z & x are originally negative.
the opposite if y<0, x will be on the right of zero and z will be on the right of x. => z>x. and both are originally positive. in both cases it seems that the equation is correct. but don't bother to check that just look at the extra data and continue from there.
If you find it hard to understand with variables just use numbers instead.
now let's look at (1) it tells you that z<x so we know that this is the first case only => the equality is definitely correct (just use numbers that maintain the data in the first case) - Sufficient.
(2) it tells you that y>0. again we are at the first case. - Sufficient.

pay attention that this is not a coincidence that the extra data given leads you in (1) and (2) to the same conclusion. if it doesn't you should suspect whether it's D or not.

Hope that helps.
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##### General Discussion
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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20 Jun 2012, 05:05
9
2
rgtiwari wrote:
If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0

You can solve such questions easily by re-stating '< 0' as 'negative' and '> 0' as 'positive'.

zy < xy < 0 implies both 'zy' and 'xy' are negative and zy is more negative i.e. has greater absolute value as compared to xy. Since y will be equal in both, z will have a greater absolute value as compared to x.

When will zy and xy both be negative? In 2 cases:
Case 1: When y is positive and z and x are both negative.
Case 2: When y is negative and z and x are both positive.

Question:
Is | x - z | + |x| = |z| ?
Is | x - z | = |z| - |x| ?
Is | z - x | = |z| - |x| ? (Since | x - z | = | z - x |)
We re-write the question only for better understanding.

Now think, when will | z - x | = |z| - |x| ? It happens when x and z have the same sign. In case both have the same sign, they get subtracted on both sides so you get the same answer. In case they have opposite signs, they get added on LHS and subtracted on RHS and hence the equality doesn't hold.

So if we can figure whether both x and z have the same sign, we can answer the question.

As we saw above, in both case 1 and case 2, x and z must have the same sign. This implies that the equality must hold and you don't actually need the statements to answer the question. You can answer it without the statements (this shouldn't happen in actual GMAT).
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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17 Sep 2012, 15:16
2
successstory wrote:
If zy<xy<0 is |x-z| + |x| = |z|?

1. z<x>0
2. y>0

why not a?

The question surprisingly does not require either statements for it to be true.

|x-z| + |x| = |z| can be rearranged as |x-z| = |z|-|x|. Now since we have an equal sign( opposed to an inequality, in which we would have to check both sides to make sure they are both positive or both negative to square (and flip)), we square both sides to get (|x-z|)^2 = (|z|-|x|)^2 which yields ===> xz=|x||z|

So our question becomes: Is xz=|x||z|?

Well this can only be true if (x>0 and z>0) OR (x<0 and z<0). In other words x and z must both be the same sign for the above statement to be true.

Now we are given zy<xy<0 as a fact. Two cases arise (+)(-)<(+)(-)<0 or (-)(+)<(-)(+)<0. Notice in both cases x and z are always the same sign!!!!! This statement is true from the gecko. Hence whatever unnecessary statement GMAC tells us will be sufficient. This is an usual problem you wont see again.
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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18 Sep 2012, 00:44
2
rgtiwari wrote:
If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0

Since zy < 0 and xy < 0, both z and x have opposite sign to y, so they must be either both positive or both negative. In other words, we know that xz > 0.

(1) Given that z < x, when both z and x are negative, |z - x| + |x| = -z + x + (-x) = -z = |z| TRUE
z and x cannot be both positive, because then y would be negative, and from zy < xy we would obtain that z > x.
Sufficient.

(2) Knowing that y > 0, we can deduce that both z and x are negative. In addition, from zy < xy it follows that z < x, and we are in the same case as above.
Sufficient.

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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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20 Dec 2012, 09:32
1
VeritasPrepKarishma wrote:
Now think, when will | z - x | = |z| - |x| ? It happens when x and z have the same sign.

If | z - x |= |z| - |x| assuming x & z have same sign
| x - z |= |z| - |x| ; Since | x - z | = | z - x |

Implies |z| - |x| = |x| - |z|
which is defn not true
z = 1 , x = 2
Plugging, we get
- 1 = 1 ?!?

Have I misunderstood something?
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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20 Dec 2012, 20:50
2
eaakbari wrote:

If | z - x |= |z| - |x| assuming x & z have same sign
| x - z |= |z| - |x| ; Since | x - z | = | z - x |

Implies |z| - |x| = |x| - |z|
which is defn not true
z = 1 , x = 2
Plugging, we get
- 1 = 1 ?!?

Have I misunderstood something?

Before discussing this part, I have discussed in my post that absolute value of z must be greater than absolute value of x.
If absolute value of z is not greater than absolute value of x, then | z - x |= |z| - |x| does not hold when x and z have the same sign.

Since |z| must be greater than |x|,
| x - z | = |x| - |z| does not hold.
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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03 Dec 2015, 05:54
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0

We get y(z-x)<0, is |x-z|=|z|-|x|?, zx>0 and z>x? if we modify the question.
There are 3 variables (x,y,z) and 1 equation in the original condition,
2 more equations in the given conditions, so there is high chance (C) will be the answer.
But condition 1=condition 2 answering the question 'no' and sufficient.

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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07 Apr 2016, 08:15
Experts please help. What level would you rate this problem? I am currently doing a DS set and got this question right in around 2 minutes but then read that its a sub-600 level some where. Please confirm?
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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07 Apr 2016, 08:22
AtharvMankotia wrote:
Experts please help. What level would you rate this problem? I am currently doing a DS set and got this question right in around 2 minutes but then read that its a sub-600 level some where. Please confirm?

Hi,

few points--
1) Here it is marked 700 level Q....
2) In actuality it should be close to 700 only sice it deals in two difficult and confusing topics for many - Modulus and Inequalities

Having said that Do not worry about the difficulty level ans least of all, do not let it effect your frame of mind.., it varies from person to person and their strengths. Some one good at inequality may feel it is very easy and at the same time may find a simple Probability as a uphill task...
There are few who are able to do 600-700 level but falter on sub-600, as they look for some trap everywhere..
So, Do not worry much about all this and master all topics
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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14 Jun 2016, 05:38
Bunuel wrote:
This is not a good question, (well at least strange enough) as neither of statement is needed to answer the question, stem is enough to do so. This is the only question from the official source where the statements aren't needed to answer the question. I doubt that such a question will occur on real test but if it ever happens then the answer would be D.

If $$zy<xy<0$$ is $$|x-z|+|x| = |z|$$

Look at the inequality $$zy<xy<0$$:

We can have two cases:

A. If $$y<0$$ --> when reducing we should flip signs and we'll get: $$z>x>0$$.
In this case: as $$z>x$$ --> $$|x-z|=-x+z$$; as $$x>0$$ and $$z>0$$ --> $$|x|=x$$ and $$|z|=z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$-x+z+x=z$$ --> $$0=0$$, which is true.

And:

B. If $$y>0$$ --> when reducing we'll get: $$z<x<0$$.
In this case: as $$z<x$$ --> $$|x-z|=x-z$$; as $$x<0$$ and $$z<0$$ --> $$|x|=-x$$ and $$|z|=-z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$x-z-x=-z$$ --> $$0=0$$, which is true.

So knowing that $$zy<xy<0$$ is true, we can conclude that $$|x-z|+|x| = |z|$$ will also be true. Answer should be D even not considering the statements themselves.

As for the statements:

Statement (1) says that $$z<x$$, hence we have case B.

Statement (2) says that $$y>0$$, again we have case B.

Hope it helps.

Hi Bunuel,

In step A, How did you get the below:
|x−z|=−x+z
Why did you take negative sign on removing the Modulus?
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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14 Jun 2016, 06:59
nishatfarhat87 wrote:
Bunuel wrote:
This is not a good question, (well at least strange enough) as neither of statement is needed to answer the question, stem is enough to do so. This is the only question from the official source where the statements aren't needed to answer the question. I doubt that such a question will occur on real test but if it ever happens then the answer would be D.

If $$zy<xy<0$$ is $$|x-z|+|x| = |z|$$

Look at the inequality $$zy<xy<0$$:

We can have two cases:

A. If $$y<0$$ --> when reducing we should flip signs and we'll get: $$z>x>0$$.
In this case: as $$z>x$$ --> $$|x-z|=-x+z$$; as $$x>0$$ and $$z>0$$ --> $$|x|=x$$ and $$|z|=z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$-x+z+x=z$$ --> $$0=0$$, which is true.

And:

B. If $$y>0$$ --> when reducing we'll get: $$z<x<0$$.
In this case: as $$z<x$$ --> $$|x-z|=x-z$$; as $$x<0$$ and $$z<0$$ --> $$|x|=-x$$ and $$|z|=-z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$x-z-x=-z$$ --> $$0=0$$, which is true.

So knowing that $$zy<xy<0$$ is true, we can conclude that $$|x-z|+|x| = |z|$$ will also be true. Answer should be D even not considering the statements themselves.

As for the statements:

Statement (1) says that $$z<x$$, hence we have case B.

Statement (2) says that $$y>0$$, again we have case B.

Hope it helps.

Hi Bunuel,

In step A, How did you get the below:
|x−z|=−x+z
Why did you take negative sign on removing the Modulus?

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: $$|5|=5$$.

So, to answer your question: we have that $$z>x$$, which is the same as $$x-z<0$$, thus $$|x-z|=-(x-z)=-x+z$$.

Hope it's clear.
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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31 Oct 2016, 13:38
VeritasPrepKarishma wrote:
rgtiwari wrote:
If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0

You can solve such questions easily by re-stating '< 0' as 'negative' and '> 0' as 'positive'.

zy < xy < 0 implies both 'zy' and 'xy' are negative and zy is more negative i.e. has greater absolute value as compared to xy. Since y will be equal in both, z will have a greater absolute value as compared to x.

When will zy and xy both be negative? In 2 cases:
Case 1: When y is positive and z and x are both negative.
Case 2: When y is negative and z and x are both positive.

Question:
Is | x - z | + |x| = |z| ?
Is | x - z | = |z| - |x| ?
Is | z - x | = |z| - |x| ? (Since | x - z | = | z - x |)
We re-write the question only for better understanding.

Now think, when will | z - x | = |z| - |x| ? It happens when x and z have the same sign. In case both have the same sign, they get subtracted on both sides so you get the same answer. In case they have opposite signs, they get added on LHS and subtracted on RHS and hence the equality doesn't hold.

So if we can figure whether both x and z have the same sign, we can answer the question.

As we saw above, in both case 1 and case 2, x and z must have the same sign. This implies that the equality must hold and you don't actually need the statements to answer the question. You can answer it without the statements (this shouldn't happen in actual GMAT).

but for the equality to hold true done we need to know for sure that /z/ > /x/ and that they have the same sign??
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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01 Nov 2016, 02:27
yezz wrote:
VeritasPrepKarishma wrote:
rgtiwari wrote:
If zy < xy < 0, is | x - z | + |x| = |z|?

(1) z < x
(2) y > 0

You can solve such questions easily by re-stating '< 0' as 'negative' and '> 0' as 'positive'.

zy < xy < 0 implies both 'zy' and 'xy' are negative and zy is more negative i.e. has greater absolute value as compared to xy. Since y will be equal in both, z will have a greater absolute value as compared to x.

When will zy and xy both be negative? In 2 cases:
Case 1: When y is positive and z and x are both negative.
Case 2: When y is negative and z and x are both positive.

Question:
Is | x - z | + |x| = |z| ?
Is | x - z | = |z| - |x| ?
Is | z - x | = |z| - |x| ? (Since | x - z | = | z - x |)
We re-write the question only for better understanding.

Now think, when will | z - x | = |z| - |x| ? It happens when x and z have the same sign. In case both have the same sign, they get subtracted on both sides so you get the same answer. In case they have opposite signs, they get added on LHS and subtracted on RHS and hence the equality doesn't hold.

So if we can figure whether both x and z have the same sign, we can answer the question.

As we saw above, in both case 1 and case 2, x and z must have the same sign. This implies that the equality must hold and you don't actually need the statements to answer the question. You can answer it without the statements (this shouldn't happen in actual GMAT).

but for the equality to hold true done we need to know for sure that /z/ > /x/ and that they have the same sign??

Both points have been considered.

The first sentence of the solution states:
"zy < xy < 0 implies both 'zy' and 'xy' are negative and zy is more negative i.e. has greater absolute value as compared to xy. Since y will be equal in both, z will have a greater absolute value as compared to x."

Since |zy| > |xy|,
|z||y| > |x||y|
Divide by |y| on both sides
|z| > |x|

Also, note the other highlighted sentence:
"Now think, when will | z - x | = |z| - |x| ? It happens when x and z have the same sign."
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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25 Feb 2017, 13:11
Bunuel wrote:
This is not a good question, (well at least strange enough) as neither of statement is needed to answer the question, stem is enough to do so. This is the only question from the official source where the statements aren't needed to answer the question. I doubt that such a question will occur on real test but if it ever happens then the answer would be D.

If $$zy<xy<0$$ is $$|x-z|+|x| = |z|$$

Look at the inequality $$zy<xy<0$$:

We can have two cases:

A. If $$y<0$$ --> when reducing we should flip signs and we'll get: $$z>x>0$$.
In this case: as $$z>x$$ --> $$|x-z|=-x+z$$; as $$x>0$$ and $$z>0$$ --> $$|x|=x$$ and $$|z|=z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$-x+z+x=z$$ --> $$0=0$$, which is true.

And:

B. If $$y>0$$ --> when reducing we'll get: $$z<x<0$$.
In this case: as $$z<x$$ --> $$|x-z|=x-z$$; as $$x<0$$ and $$z<0$$ --> $$|x|=-x$$ and $$|z|=-z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$x-z-x=-z$$ --> $$0=0$$, which is true.

So knowing that $$zy<xy<0$$ is true, we can conclude that $$|x-z|+|x| = |z|$$ will also be true. Answer should be D even not considering the statements themselves.

As for the statements:

Statement (1) says that $$z<x$$, hence we have case B.

Statement (2) says that $$y>0$$, again we have case B.

Hope it helps.

why in case A, have you expanded |x-z| as -x+z, but |x| and |y| as x and y?? shouldnt |x| and |y| be expanded as -x and -y as well?
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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26 Feb 2017, 03:22
1
OreoShake wrote:
Bunuel wrote:
This is not a good question, (well at least strange enough) as neither of statement is needed to answer the question, stem is enough to do so. This is the only question from the official source where the statements aren't needed to answer the question. I doubt that such a question will occur on real test but if it ever happens then the answer would be D.

If $$zy<xy<0$$ is $$|x-z|+|x| = |z|$$

Look at the inequality $$zy<xy<0$$:

We can have two cases:

A. If $$y<0$$ --> when reducing we should flip signs and we'll get: $$z>x>0$$.
In this case: as $$z>x$$ --> $$|x-z|=-x+z$$; as $$x>0$$ and $$z>0$$ --> $$|x|=x$$ and $$|z|=z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$-x+z+x=z$$ --> $$0=0$$, which is true.

And:

B. If $$y>0$$ --> when reducing we'll get: $$z<x<0$$.
In this case: as $$z<x$$ --> $$|x-z|=x-z$$; as $$x<0$$ and $$z<0$$ --> $$|x|=-x$$ and $$|z|=-z$$.

Hence in this case $$|x-z|+|x|=|z|$$ will expand as follows: $$x-z-x=-z$$ --> $$0=0$$, which is true.

So knowing that $$zy<xy<0$$ is true, we can conclude that $$|x-z|+|x| = |z|$$ will also be true. Answer should be D even not considering the statements themselves.

As for the statements:

Statement (1) says that $$z<x$$, hence we have case B.

Statement (2) says that $$y>0$$, again we have case B.

Hope it helps.

why in case A, have you expanded |x-z| as -x+z, but |x| and |y| as x and y?? shouldnt |x| and |y| be expanded as -x and -y as well?

A. If $$y<0$$ --> when reducing we should flip signs and we'll get: $$z>x>0$$.

Since $$z>x$$ --> x - z < 0, thus $$|x-z|=-(x-z) =-x+z$$;
Since $$x>0$$ and $$z>0$$ --> $$|x|=x$$ and $$|z|=z$$.
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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30 Jun 2018, 16:05
HI Bunuel

I have a question. I answered the question correctly but I'm not sure if the steps I took are correct.

I looked at the equation |x-Z| + |x| = |z| and noticed that with the absolute, each term will be positive. So I squared each term.

I ended up with 2X^2 -2XZ = 0. With a little rearranging, I got X=Z. So the question is really asking if Z=X

From here I was able to answer both statements quickly.

ST1: Z<X Sufficient. Z doesn't equal X
ST2: Y>0 Sufficient. If Y is positive than Z and X are negative. The inequality lists ZY<ZX so Z doesn't equal X.

Question - Was squaring each term legit in this case because the expression has the same positive sign due to the absolute value sign?

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Posts: 47033
Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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30 Jun 2018, 23:29
1
hdavies wrote:
HI Bunuel

I have a question. I answered the question correctly but I'm not sure if the steps I took are correct.

I looked at the equation |x-Z| + |x| = |z| and noticed that with the absolute, each term will be positive. So I squared each term.

I ended up with 2X^2 -2XZ = 0. With a little rearranging, I got X=Z. So the question is really asking if Z=X

From here I was able to answer both statements quickly.

ST1: Z<X Sufficient. Z doesn't equal X
ST2: Y>0 Sufficient. If Y is positive than Z and X are negative. The inequality lists ZY<ZX so Z doesn't equal X.

Question - Was squaring each term legit in this case because the expression has the same positive sign due to the absolute value sign?

That's not correct. You cannot square individual terms the way you did. Formula of square of the sum of two numbers is $$(a + b)^2 = a^2 + 2ab + b^2$$. What I mean is that if you square $$a + b = c$$ you get $$(a + b)^2 = c^2$$ NOT $$a^2 + b^2 = c^2$$. So, if you square $$|x - z| + |x| = |z|$$, you get $$(x - z)^2 + 2*|x - z|*|x| + z^2 = z^2$$.
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Re: If zy < xy < 0, is |x - z | + |x| = |z|? [#permalink]

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01 Jul 2018, 09:25
Bunuel wrote:
hdavies wrote:
HI Bunuel

I have a question. I answered the question correctly but I'm not sure if the steps I took are correct.

I looked at the equation |x-Z| + |x| = |z| and noticed that with the absolute, each term will be positive. So I squared each term.

I ended up with 2X^2 -2XZ = 0. With a little rearranging, I got X=Z. So the question is really asking if Z=X

From here I was able to answer both statements quickly.

ST1: Z<X Sufficient. Z doesn't equal X
ST2: Y>0 Sufficient. If Y is positive than Z and X are negative. The inequality lists ZY<ZX so Z doesn't equal X.

Question - Was squaring each term legit in this case because the expression has the same positive sign due to the absolute value sign?

That's not correct. You cannot square individual terms the way you did. Formula of square of the sum of two numbers is $$(a + b)^2 = a^2 + 2ab + b^2$$. What I mean is that if you square $$a + b = c$$ you get $$(a + b)^2 = c^2$$ NOT $$a^2 + b^2 = c^2$$. So, if you square $$|x - z| + |x| = |z|$$, you get $$(x - z)^2 + 2*|x - z|*|x| + z^2 = z^2$$.

Thank you
Re: If zy < xy < 0, is |x - z | + |x| = |z|?   [#permalink] 01 Jul 2018, 09:25
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