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

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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

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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 8358 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

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

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