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Manager  Joined: 17 Nov 2009
Posts: 219
Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 71% (01:08) correct 29% (01:12) wrong based on 278 sessions

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Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10
Math Expert V
Joined: 02 Sep 2009
Posts: 53800

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agnok wrote:
Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10

Is $$|x-3|<7$$?

Let's see for which range(s) of $$x$$ this inequality holds true. One check point $$x=3$$ (check point the value of $$x$$ when the expression in || equals to zero), so we should check two ranges:

When $$x<3$$ --> $$|x-3|$$ becomes $$-x+3$$ --> $$-x+3<7$$ --> $$-4<x$$ --> $$-4<x<3$$;
When $$x\geq{3}$$ --> $$|x-3|$$ becomes $$x-3$$ --> $$x-3<7$$ --> $$x<10$$ --> $$3\leq{x}<10$$;

So inequality $$|x-3|<7$$ holds true for $$-4<x<10$$.

(1) $$x>0$$. Not sufficient.
(2) $$x<10$$. Not sufficient.

(1)+(2) $$0<x<10$$ --> we know that in this range inequality $$|x-3|<7$$ holds true (this range is the subset of the range we've got in the beginning). Sufficient.

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You would have read in theory of mods that mod is nothing but the distance from x = 0 on the number line.
This means that if |x| = 4, x is a point at a distance of 4 units from 0. So x could be 4 or -4 (which suits our understanding of mods)

Now, |x – 3| is the distance from the point 3 on the number line. So if I say
|x – 3| = 7, I am looking for points which are at a distance of 7 from point 3. These points will be 10 and -4. These are the solutions of x in this equation.

Coming to our question, |x – 3| < 7 means we are looking for points whose distance from 3 is less than 7. There will be many such points e.g. 4, 5, 6, -2, -1 etc that satisfy our inequality as is apparent from the diagram below:
Attachment: Ques.jpg [ 5.28 KiB | Viewed 4068 times ]

Points beyond 10 on the right side of the number line and points beyond -4 on the left side of the number line will have a distance of more than 7 and hence, do not satisfy our inequality.
Statement I: x > 0
There are points greater than 0 that satisfy our inequality (e.g. 1, 5, 7 etc) and there are those that do not satisfy our inequality (e.g. 11, 12, 18 etc). Hence this statement is not sufficient.
Statement II: x < 10
Again, using the same logic as above, this statement is not sufficient.

When we combine the two statements, we see that all the points satisfying 0 < x < 10, satisfy our inequality. Hence we can say 'Yes, |x – 3| is less than 7' and we get (C) as our answer.
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Intern  Joined: 23 Dec 2011
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GMAT 1: 620 Q47 V28 Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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2
Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10

Need explanation please. Thank you!
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Posts: 36
Schools: HBS '15 (M)
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Re: Is |x – 3| < 7 ?  [#permalink]

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2
First, you need to simplify the stem. Since there is an absolute value in the expression, remove the absolute value signs and setup two inequalities.

A. x-3 < 7 ?
x < 10 ?

B. -(x-3) < 7 ?
-x+3 < 7 ?
-x < 4 ?
x > 4 ?

Next, combine the two inequalities: -4 < x < 10 ? In other words, the question is asking whether x is between -4 and 10.

Now, evaluate the statements.

(1) x > 0
INSUFFICIENT: because x>0, x>-4. However, the statement does not prove that x<10. For example, if x = 8, the answer to the question is yes. If x = 15, the answer to the question is no.

(2) x < 10
INSUFFICIENT: this statment indicates that x<10. However, the statement does not prove that x>-4. For example, if x = 8, the answer to the question is yes. If x = -10, the answer to the question is no.

Finally, combine the two statements. Combined, the two statements say: 0<x<10. If x is between 0 and 10, x must also be between -4 and 10. Therefore, the answer to the question is C.

Math Expert V
Joined: 02 Sep 2009
Posts: 53800
Re: Is |x – 3| < 7 ?  [#permalink]

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1
One can also use distance concept to solve this questions, which for this particular question would be an easier approach.

Is |x-3| < 7 ?

|x-3| is just the distance between 3 and x on the number line. The question basically asks whether this distance is less than 7: --(-4)------(3)------(10)-- so, whether -4<x<10 is true?

(1) x > 0. Not sufficient.
(2) x < 10. Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is 0<x<10, so the answer to the question is YES. Sufficient.

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Re: Is |x – 3| < 7 ?  [#permalink]

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DagwoodDeluxe wrote:
First, you need to simplify the stem. Since there is an absolute value in the expression, remove the absolute value signs and setup two inequalities.

A. x-3 < 7 ?
x < 10 ?

B. -(x-3) < 7 ?
-x+3 < 7 ?
-x < 4 ?
x > 4 ?

Next, combine the two inequalities: -4 < x < 10 ? In other words, the question is asking whether x is between -4 and 10.

Is |x – 3| < 7 ?

x-3 < 7
x < 10
OR
-x+3 < 7
x > -4

When we consider both inequalities, question becomes -

is x < 10
OR
is x > -4

Isn't it wrong to combine inequalities like this -4 < x < 10 and consider two scenarios as AND scenarios?
Math Expert V
Joined: 02 Sep 2009
Posts: 53800
Re: Is |x – 3| < 7 ?  [#permalink]

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Apex231 wrote:
Is |x – 3| < 7 ?

x-3 < 7
x < 10
OR
-x+3 < 7
x > -4

When we consider both inequalities, question becomes -

is x < 10
OR
is x > -4

Isn't it wrong to combine inequalities like this -4 < x < 10 and consider two scenarios as AND scenarios?

"Is |x-3|<7?" does mean "is -4<x<10?" Refer to my solution above.

Or another way: if x<3, then -(x-3)<7 --> x>-4 AND if x>=3, then x-3<7 --> x<10. So, |x-3|<7 to holds true for -4<x<10. So, we don't have 2 separate inequalities: x>-4 OR x<10, we have 2 cases and both must hold true for |x-3|<7 to hold true.

Hope it's clear.
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Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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In the post below we considered two scenarios as OR scenarios and in this question we are considering them as AND scenarios. I understand the distance approach but still little confused. How is this question different from one below? Please help!

http://gmatclub.com/forum/ds-inequalities-126369.html

In the question below we considered two cases separately and didn't combine i.e. didn't do y + 3 <= x <= 3 - y

If y >= 0, What is the value of x?
(1) |x-3| >= y
(2) |x-3| <= -y

1) |x-3| >= y

x -3 >=y
x >= y + 3

OR

3 - x >=y
x <= 3 - y

Posted from my mobile device
Math Expert V
Joined: 02 Sep 2009
Posts: 53800
Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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Apex231 wrote:
In the post below we considered two scenarios as OR scenarios and in this question we are considering them as AND scenarios. I understand the distance approach but still little confused. How is this question different from one below? Please help!

Forget about or, and and y. The point was that you were combining two cases into one when you couldn't do that.

Consider this: |x-3|>1
If x<=3 --> -(x-3)>1 --> x<2;
If x>3 --> (x-3)>1 --> x>4.

And then you were writing 4<x<2, which is obviously wrong.

Another example: |x-3|<1
If x<=3 --> -(x-3)<1 --> 2<x --> 2<x<=3;
If x>3 --> (x-3)>1 --> x<4. --> 3<x<4;
Now you can write 2<x<4 as it would be right.

Hope it's clear.
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Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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Thanks Bunuel. I think i understand now.

So in the first example we can combine ranges because it's a continuous range but in the second example the range is not continuous so we can't combine.
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Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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if X>0 , then from 1 - 9 , the inequality is correct and if X>10 it doesnt hold correct -- 1 Not sufficient
if X<10, then from 9 - (-3) , the inequality is correct and for X< -3 it doent hold correct --- 2 Not sufficient

1 n 2 togethe 0<X<10 for all the values in this range the inequality holds good.

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Math Expert V
Joined: 02 Sep 2009
Posts: 53800
Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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1
Bumping for review and further discussion.
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GRE 1: Q169 V154 Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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Here using the rule if |x|<n => x=> (-n,n)
the question wants to know if x=> (-4,10)
clearly C is sufficient...
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Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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Hi Bunuel
Question is asking about whether -4<x<10 so Sts. 1+2 give us 0<x>10 so how we consider it is true. it is not asking whether x is positive and < 10.
I need to get the missing point please.
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GMAT 1: 590 Q49 V20 GMAT 2: 730 Q50 V38 Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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DesecratoR wrote:
Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10

Need explanation please. Thank you!

we have to find the value of lx-3l<7
simplify this statement
-7<(x-3)<7
on solving this equation we get
-4<x<10
so basically we have to find whether the of x lies betwee -4 and 10

Statement 1
x>0
so from this we can say the value of x is greater than 0 and can be any positive value fro example 50
so A is not possible
Statement 2
x<10
similarly from this we only know value of x< 10 and can even -50
so B is also not possible

on combining both statement we can conclude 0<x<10
and after combining we get that the vaue of x is in between -4<x<10
So C is ans
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GMAT 1: 590 Q49 V20 GMAT 2: 730 Q50 V38 Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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DesecratoR wrote:
Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10

Need explanation please. Thank you!

we have to find the value of lx-3l<7
simplify this statement
-7<(x-3)<7
on solving this equation we get
-4<x<10
so basically we have to find whether the of x lies betwee -4 and 10

Statement 1
x>0
so from this we can say the value of x is greater than 0 and can be any positive value fro example 50
so A is not possible
Statement 2
x<10
similarly from this we only know value of x< 10 and can even -50
so B is also not possible

on combining both statement we can conclude 0<x<10
and after combining we get that the vaue of x is in between -4<x<10
So C is ans

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GMAT 1: 720 Q49 V40 Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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DesecratoR wrote:
Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10

Need explanation please. Thank you!

Hi Bunuel and bb
This question is also seen in another thread. Can you merge this topics, please?
https://gmatclub.com/forum/is-x-3-7-1-x ... fl=similar
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GMAT 1: 720 Q49 V40 Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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agnok wrote:
Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10

The question stem says:
Is -4<x<10?
In number line, it is green part.
<----(-5)----(-4)---(-3)---(-2)---(-1)---0----1---2---3---4---5---6---7---8---9---10----11>
Statement 1:
x>0
here, x may be any positive values. This may be in the green part or may be next after the green part.
So, insufficient.
Statement 2:
x<10
Here, x may be into the green part or it may be in the red part. So, the answer may be YES or NO. So, insufficient.
Statement 1+2:
It combined says:
0<x<10
So, sufficient from the number line. The correct choice is C.
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Posts: 53800
Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10  [#permalink]

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iMyself wrote:
DesecratoR wrote:
Is |x – 3| < 7 ?

(1) x > 0
(2) x < 10

Need explanation please. Thank you!

Hi Bunuel and bb
This question is also seen in another thread. Can you merge this topics, please?
https://gmatclub.com/forum/is-x-3-7-1-x ... fl=similar

Thank you for noticing it. Merged the topics.
_________________ Re: Is |x – 3| < 7 ? (1) x > 0 (2) x < 10   [#permalink] 03 Feb 2017, 03:08

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