How many positive integers less than 10 satisfy ||x - 3| - 4| 3?

Question

How many positive integers less than 10 satisfy  ||x - 3| - 4| > 3 ?

A. 0
B. 1
C. 2
D. 3
E. 4

Are You Up For the Challenge: 700 Level Questions

GMATWhizTeam · Accepted Answer

Solution • Let us assume that |x – 3| = z . o Then, | z – 4| > 3 o ⟹ Either (z – 4) > 3 or (z -4) < -3 o ⟹ z > 7 or z < 1 • Therefore, we have |x -3| < 1 and |x – 3| > 7. Let’s solve these inequalities one by one. o |x – 3| < 1 ⟹ -1 < (x -3) < 1 ⟹2 < x < 4  Thus, positive integral value of x = 3 (i.e. one positive integral soluion) o |x – 3| > 7 ⟹ Either (x – 3) > 7 or (x – 3) < -7 ⟹ x > 10 or x < -4  So, no solution in this case. • So, only one positive integer (x = 3) which is less than 10 satisfies the given equation. Thus, the correct answer is Option B.

yashikaaggarwal · Answer

IMO is B (1)

Since x is a positive integer.
X=9,8,7,6,5,4,3,2&1
Now.
Since -7 is in subtraction.
||9-3|-4|=2<3
||8-3|-4|=1<3
||7-3|-4|=0<3
||6-3|-4|=1<3
||5-3|-4|=2<3
||4-3|-4|=3=3
||3-3|-4|=4>3 (only integral value satisfying equation)
||2-3|-4|=3=3
||1-3|-4|=2<3

IMO is B

Posted from my mobile device

nick1816 · Answer

Curve in red represents y=|x - 3| - 4| Untitled.png From the curve we can see that y is greater than 3 at x=3 only, where 0

SiffyB · Answer

Hi  nick1816 ,

Can you please explain how to draw this graph from inequalities?

Thanks in advance!!

rishab0507 · Answer

One way is to test all values from 0-9, Will be bit lengthy but will give answer, But this technique will not be of use if we are given less than 20 or 30. So we need to understand solution

||x−3|−4|>3
Take |x-3| = y , Now we have
|y-4| >3, For this solution to satisfy we need to have Y >7, 
now we reach to equation |x-3 | >7 for how many values of X, less than 10. Which is ZERO as minimum value that can satisfy is 11.
Till now 0 values. But we need to test 0 also, because if y= 0 or x-3 =0, then equation is satisfied. 
x- 3 =0 => x=3, 
Thus only 1 value satisfy equation i.e 3

IMO B

Plz give Kudos to Support

nick1816 · Answer

abcdddddd
||x−3|−4|>3 represents the region in which y=|x - 3| - 4| lies above the line y=3

Attachment:

Untitled.png [ 44.4 KiB | Viewed 8752 times ]

It's easy to find the intersection points, since the slope of each portion of y=|x - 3| - 4| is 1 or -1 (coefficient of x is |1|) . So you have to add 1 unit in y-coordinates, if you add 1 in x-coordinate (if slope is 1) and you have to deduct 1 unit in y-coordinates, if you add 1 in x-coordinate (if slope is -1).

If you still have any doubt, you can ask.

GMATinsight · Answer

A HARD problem explained in easy way... Check video explanation to find the tricky area https://gmatclub.com/forum/download/file.php?mode=view&id=54649 https://www.youtube.com/watch?v=ZiO1Clw9zhc Screenshot 2020-05-21 at 11.09.33 AM.png

sambitspm · Answer

See the attachment

SiffyB · Answer

That was so helpful! Thank you, nick1816 .

Can you please quote some more questions where this method can be applied? Or more sources to read up from. I don't know the name of this method, so i'm not able to look up for questions that employ it.

Kinshook · Answer

Asked: How many positive integers less than 10 satisfy ||x - 3| - 4| > 3 ? ||x - 3| - 4| > 3 x = 1; |2-4| = 2 < 3: NO x =2; |1-4| = 3 = 3; NO x =3; |0-4| =4 > 3; YES x = 4; |1-4| = 3=3: NO x =5; |2-4| = 2 < 3; NO x =6; |3-4| =1 < 3; NO x = 7; |4-4| = 0<3: NO x =8; |5-4| = 1 < 3; NO x =9; |6-4| =2 < 3; NO IMO B

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How many positive integers less than 10 satisfy ||x - 3| - 4| > 3?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions