December 09, 2018 December 09, 2018 07:00 AM PST 09:00 AM PST Attend this Free Algebra Webinar and learn how to master Inequalities and Absolute Value problems on GMAT. December 10, 2018 December 10, 2018 10:00 PM PST 11:00 PM PST Practice the one most important Quant section  Integer properties, and rapidly improve your skills.
Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 51035

For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
09 Jan 2017, 06:03
Question Stats:
52% (02:30) correct 48% (02:18) wrong based on 725 sessions
HideShow timer Statistics




Math Expert
Joined: 02 Aug 2009
Posts: 7095

For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
09 Jan 2017, 06:54
Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite Hi sobby, The actual way would be... Since both sides are positive, square both sides \((x6)^2>(3x+6)^2..........x^212x+36>9x^2+36x+36.........8x^2+48x<0\) 8x(x+6)<0..... For 8x(x+6) to be NEGATIVE, one of 8x and x+6 will be negative and other positive.. If x is positive, 8x will be positive and x+6 will also be positive.. So x should be NEGATIVE..Then 8x will be negative, and thus x+6 should be positive .. For that \(x+6>0.....X>6....\) But x<0. So range becomes \(0>x>6\)... Values are \(1,2,3,4,5\) 5 value C
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor




Senior CR Moderator
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1376
Location: Viet Nam

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
09 Jan 2017, 07:11
Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite In this question, in my opinion, there are 2 possible solutions. Solution 1.\(x – 6 > 3x + 6 \implies x – 6  3x + 6>0\) If \(x<2\) we have \((6x)(3x6)=6x+3x+6=2x+12 > 0 \implies x >6\). Hence \(6 <x <2\) If \(2 \leq x < 6\) we have \((6x)(3x+6)=4x>0 \implies x<0\). Hence \(2 \leq x < 0\) If \(x \geq 6\) we have \((x6)(3x+6)=2x12>0 \implies x<6\). There is no satisfied value of \(x\) in this case. Combine all cases we have \(6 < x < 0\). Thus, \(x\) could receive 5 integer value \(\{5;4;3;2;1\}\) The answer is C. Solution 2.\(x – 6 > 3x + 6 \implies (x6)^2 > (3x+6)^2\) \(\implies x^212x+36 > 9x^2+36x+36 \implies 8x^2+48x<0\) \(\implies 8x(x+6)<0 \implies 6<x<0\).
_________________
Actual LSAT CR bank by Broall
How to solve quadratic equations  Factor quadratic equations Factor table with sign: The useful tool to solve polynomial inequalities Applying AMGM inequality into finding extreme/absolute value
New Error Log with Timer




Director
Joined: 14 Nov 2014
Posts: 635
Location: India
GPA: 3.76

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
09 Jan 2017, 06:32
Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite by substituting values i can see 5 as answer ... No positive value for x can satisfy the equation...1 x=0 can equalize the equation2 1 to 5 will satisfy the equation...and 6 will equalize the condition3 so the answer is c  5 .. is there any other way to solve i'e without doing actual substitution ... Thanks..



Director
Joined: 14 Nov 2014
Posts: 635
Location: India
GPA: 3.76

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
09 Jan 2017, 06:39
sobby wrote: Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite by substituting values i can see 5 as answer ... No positive value for x can satisfy the equation...1 x=0 can equalize the equation2 1 to 5 will satisfy the equation...and 6 will equalize the condition3 so the answer is c  5 .. is there any other way to solve i'e without doing actual substitution ... Thanks.. 2nd approach....... we can do it by squaring both side and using number line to solve it ... we will get the roots as 0 and 6 and using this we can deduce only number between 0 and 6 will satisfy the actual solution.. Bunuel : Is this approach correct I have one doubt while squaring in absolute ....what necessary condition we need to consider....



Math Expert
Joined: 02 Sep 2009
Posts: 51035

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
09 Jan 2017, 07:17
sobby wrote: sobby wrote: Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite by substituting values i can see 5 as answer ... No positive value for x can satisfy the equation...1 x=0 can equalize the equation2 1 to 5 will satisfy the equation...and 6 will equalize the condition3 so the answer is c  5 .. is there any other way to solve i'e without doing actual substitution ... Thanks.. 2nd approach....... we can do it by squaring both side and using number line to solve it ... we will get the roots as 0 and 6 and using this we can deduce only number between 0 and 6 will satisfy the actual solution.. Bunuel : Is this approach correct I have one doubt while squaring in absolute ....what necessary condition we need to consider.... We can raise both parts of an inequality to an even power if we know that both parts of an inequality are nonnegative (the same for taking an even root of both sides of an inequality). So, if you have absolute values on both sides of the inequality then you can safely square. Check for more the following topics: Inequalities Made Easy!Solving Quadratic Inequalities  Graphic ApproachInequality tipsDS Inequalities Problems PS Inequalities Problems 700+ Inequalities problemsHope it helps.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Senior Manager
Status: Countdown Begins...
Joined: 03 Jul 2016
Posts: 294
Location: India
Concentration: Technology, Strategy
GPA: 3.7
WE: Information Technology (Consulting)

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
09 Jan 2017, 20:16
chetan2u wrote: Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite Hi sobby, The actual way would be... Since both sides are positive, square both sides \((x6)^2>(3x+6)^2..........x^212x+36>9x^2+36x+36.........8x^2+36x<0\) 8x(x+6)<0..... For 8x(x+6) to be NEGATIVE, one of 8x and x+6 will be negative and other positive.. If x is positive, 8x will be positive and x+6 will also be positive.. So x should be NEGATIVE.. Then 8x will be negative, and thus x+6 should be positive .. For that x+6>0.....X>6.... But x<0. So range becomes 0>x>6... Values are 1,2,3,4,5 5 value C
Attachments
File comment: Here is another graphical approach.
The blue graph represents x6 and red graph represents 3x+6 and these two graphs meet each other at x=0 and x=6 For x6 to be greater than 3x+6 the blue graph should be above red graph. So our solution should be 6<x<0 i.e. 1,2,3,4,5.
Experts please comment if this approach is correct. Also I tried applying the same approach to 3 modulus equations but it is not so handy as it is here.. Please suggest some method for 3 modulus problems as x – 8 + 5 – x > x + 7
Graph1.PNG [ 34.52 KiB  Viewed 7205 times ]



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8649
Location: Pune, India

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
10 Jan 2017, 04:27
Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite Think in terms of the definition of absolute values: x  6 > 3*x + 2 Distance from 6 is greater than 3 times the distance from 2  (2)  (0)  (6)  At point 0, the distance from 6 will be equal to 3 times the distance from 2. So on the left of 0, the distance from 6 will be greater. The distance between 2 and 6 is 8. Distance from 6 will be equal to 3 times the distance from 2 at a point 4 units to the left of 2 i.e. 6. To its left, distance from 6 will be less. So between 6 to 0, distance from 6 is greater than 3 times the distance from 2. We have 5 integer values lying between these two. Answer (C)
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2830

For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
11 Jan 2017, 07:05
Bunuel wrote: For how many integer values of x, is \(x – 6 > 3x + 6\)?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite To solve an inequality where the absolute value of an expression is greater than (or less than) the absolute value of another expression, we need to find the values that make the absolute values of the two expressions equal to each other. That is, to solve x – 6 > 3x + 6, we first need to solve x – 6 = 3x + 6. Recall that if A = B, then either A = B (Case 1) or A = B (Case 2). Case 1: When \(x  6 = 3x + 6\): \(x  6 = 3x + 6\) \(2x = 12\) \(x = 6\) Case 2: When \(x  6 = (3x + 6)\): \(x  6 = (3x + 6)\) \(x  6 = 3x  6\) \(4x = 0\) \(x = 0\) These two values, 6 and 0, are the critical values for determining the solution to the given inequality. Now when we place these two values on the number line, we will see that they partition the number line into 3 intervals: \(x < 6\) \(6 < x < 0\) \(x > 0\) For each of these intervals, we will pick a value in that interval (for example, for x < 6, we can pick 7) and test whether it satisfies the given inequality. If it does, then EVERY number in that interval will satisfy the inequality. If it doesn’t, then NO numbers in that interval will satisfy the inequality. Recall that our given inequality is x – 6 > 3x + 6. Let’s begin testing the intervals. 1) For \(x < 6\), let’s pick x = 7: 7 – 6 > 3(7) + 6 ? \(13 > 15\)? \(13 > 15\) → No!Thus, no numbers in the interval \(x < 6\)will satisfy \(x – 6 > 3x + 6.\) 2) For \(6 < x < 0\), let’s pick x = 1: \(1 – 6 > 3(1) + 6\)? \(7 > 3\) ? \(7 > 3\)→ YesThus, every number in the interval \(6 < x < 0\) will satisfy \(x – 6 > 3x + 6.\) 3) For \(x > 0\), let’s pick x = 1: \(1 – 6 > 3(1) + 6\) ? \(5 > 9\) ? \(5 > 9\) → NoThus, no numbers in the interval \(x > 0\) will satisfy \(x – 6 > 3x + 6.\)As we can see, the only interval that satisfies \(x – 6 > 3x + 6\) is \(6 < x < 0\). In other words, the solution to \(x – 6 > 3x + 6\) is \(6 < x < 0\). In this interval, there are 5 integers: 5, 4, 3, 2, and 1. Answer: C
_________________
Jeffery Miller
Head of GMAT Instruction
GMAT Quant SelfStudy Course
500+ lessons 3000+ practice problems 800+ HD solutions



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8649
Location: Pune, India

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
27 Feb 2017, 23:47
VeritasPrepKarishma wrote: Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite Think in terms of the definition of absolute values: x  6 > 3*x + 2 Distance from 6 is greater than 3 times the distance from 2  (2)  (0)  (6)  At point 0, the distance from 6 will be equal to 3 times the distance from 2. So on the left of 0, the distance from 6 will be greater. The distance between 2 and 6 is 8. Distance from 6 will be equal to 3 times the distance from 2 at a point 4 units to the left of 2 i.e. 6. To its left, distance from 6 will be less. So between 6 to 0, distance from 6 is greater than 3 times the distance from 2. We have 5 integer values lying between these two. Answer (C) Responding to a pm: Quote: I know very well that if x  3 < 6
Distance of x from 3 is less than 6 so 3 < x < 9.
with this understood, Can you please elaborate your answer?
First consider the equation: x  6 = 3*x + 2 That is, the point where distance from 6 is equal to three times the distance from 2. There is such a point between 6 and 2. That point is x = 0. Here the distance from 6 is 6 which is equal to three times 2, the distance of 0 from 2. How do we get x = 0? Distance from 6 should be thrice the distance from 2 Distance from 6 : Distance from 2 = 3:1 Split the distance between 6 and 2 into the ratio 3:1. Since there are 8 units between 6 and 2, you get that x is 6 units away from 6 and hence at 0. For the inequality, we want the distance from 6 to be greater than the distance from 2 so we move to the left (closer to 2). The distance from 6 will keep increasing till we reach 2 when the distance from 6 is 8. Note that now also distance from 6 is greater than 3 times the distance from 2. Now as we go further to the left, the distance from 6 keeps increasing but so does the distance from 2. There will be some point again where distance from 6 will be equal to 3 times the distance from 2. At this point, the distance of 8 units between 6 and 2 will make up twice the distance of x from 2 (x)......................(2).................(0)...................................................(6)................... < Distance from 2> <Distance from 6 > So x must be 4 units to the left of 2. So at x = 6, again, distance from 6 will be three times the distance from 2. Hence, for our inequality, x can take values between 6 and 0. This gives us 5 integer values.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Senior Manager
Status: Countdown Begins...
Joined: 03 Jul 2016
Posts: 294
Location: India
Concentration: Technology, Strategy
GPA: 3.7
WE: Information Technology (Consulting)

For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
03 Mar 2017, 19:09
VeritasPrepKarishma wrote: Responding to a pm: Quote: I know very well that if x  3 < 6
Distance of x from 3 is less than 6 so 3 < x < 9.
with this understood, Can you please elaborate your answer?
First consider the equation: x  6 = 3*x + 2 That is, the point where distance from 6 is equal to three times the distance from 2. There is such a point between 6 and 2. That point is x = 0. Here the distance from 6 is 6 which is equal to three times 2, the distance of 0 from 2. How do we get x = 0? Distance from 6 should be thrice the distance from 2 Distance from 6 : Distance from 2 = 3:1 Split the distance between 6 and 2 into the ratio 3:1. Since there are 8 units between 6 and 2, you get that x is 6 units away from 6 and hence at 0. For the inequality, we want the distance from 6 to be greater than the distance from 2 so we move to the left (closer to 2). The distance from 6 will keep increasing till we reach 2 when the distance from 6 is 8. Note that now also distance from 6 is greater than 3 times the distance from 2. Now as we go further to the left, the distance from 6 keeps increasing but so does the distance from 2. There will be some point again where distance from 6 will be equal to 3 times the distance from 2. At this point, the distance of 8 units between 6 and 2 will make up twice the distance of x from 2 (x)......................(2).................(0)...................................................(6)................... < Distance from 2> <Distance from 6 > So x must be 4 units to the left of 2. So at x = 6, again, distance from 6 will be three times the distance from 2. Hence, for our inequality, x can take values between 6 and 0. This gives us 5 integer values. Thank you so much VeritasPrepKarishma for the explanation. +1 A good thoughtful explanation. It took a little time for me to understand but I guess I need to practice more to internalize it. Just one question here, for the below inequality  x6 > x+2 can we say that the solution for this is x<2?



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8649
Location: Pune, India

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
03 Mar 2017, 22:55
RMD007 wrote: VeritasPrepKarishma wrote: Responding to a pm: Quote: I know very well that if x  3 < 6
Distance of x from 3 is less than 6 so 3 < x < 9.
with this understood, Can you please elaborate your answer?
First consider the equation: x  6 = 3*x + 2 That is, the point where distance from 6 is equal to three times the distance from 2. There is such a point between 6 and 2. That point is x = 0. Here the distance from 6 is 6 which is equal to three times 2, the distance of 0 from 2. How do we get x = 0? Distance from 6 should be thrice the distance from 2 Distance from 6 : Distance from 2 = 3:1 Split the distance between 6 and 2 into the ratio 3:1. Since there are 8 units between 6 and 2, you get that x is 6 units away from 6 and hence at 0. For the inequality, we want the distance from 6 to be greater than the distance from 2 so we move to the left (closer to 2). The distance from 6 will keep increasing till we reach 2 when the distance from 6 is 8. Note that now also distance from 6 is greater than 3 times the distance from 2. Now as we go further to the left, the distance from 6 keeps increasing but so does the distance from 2. There will be some point again where distance from 6 will be equal to 3 times the distance from 2. At this point, the distance of 8 units between 6 and 2 will make up twice the distance of x from 2 (x)......................(2).................(0)...................................................(6)................... < Distance from 2> <Distance from 6 > So x must be 4 units to the left of 2. So at x = 6, again, distance from 6 will be three times the distance from 2. Hence, for our inequality, x can take values between 6 and 0. This gives us 5 integer values. Thank you so much VeritasPrepKarishma for the explanation. +1 A good thoughtful explanation. It took a little time for me to understand but I guess I need to practice more to internalize it. Just one question here, for the below inequality  x6 > x+2 can we say that the solution for this is x<2? Yes, that is correct. x6 > x+2 The distance of x from 6 is more than the distance of x from 2. The two distances will be equal at the mid point that is at x = 2. As you move the left, the distance of x from 6 will keep increasing. To the left of 2 too, the distance from 6 will always be more than the distance from 2. Hence answer will be x < 2.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Intern
Joined: 22 Nov 2016
Posts: 13

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
24 Mar 2017, 22:09
In this question, in my opinion, there are 2 possible solutions.
Solution 1.
x–6>3x+6⟹x–6−3x+6>0x–6>3x+6⟹x–6−3x+6>0
If x<−2x<−2 we have (6−x)−(−3x−6)=6−x+3x+6=2x+12>0⟹x>−6(6−x)−(−3x−6)=6−x+3x+6=2x+12>0⟹x>−6. Hence −6<x<−2−6<x<−2
If −2≤x<6−2≤x<6 we have (6−x)−(3x+6)=−4x>0⟹x<0(6−x)−(3x+6)=−4x>0⟹x<0. Hence −2≤x<0−2≤x<0
If x≥6x≥6 we have (x−6)−(3x+6)=−2x−12>0⟹x<−6(x−6)−(3x+6)=−2x−12>0⟹x<−6. There is no satisfied value of xx in this case.
Combine all cases we have −6<x<0−6<x<0.
Thus, xx could receive 5 integer value {−5;−4;−3;−2;−1}{−5;−4;−3;−2;−1}
The answer is C.
Solution 2.
x–6>3x+6⟹(x−6)2>(3x+6)2x–6>3x+6⟹(x−6)2>(3x+6)2
⟹x2−12x+36>9x2+36x+36⟹8x2+48x<0⟹x2−12x+36>9x2+36x+36⟹8x2+48x<0
⟹8x(x+6)<0⟹−6<x<0⟹8x(x+6)<0⟹−6<x<0.
hi nguyendinhtuong . could u please explain why x<6 has no satisfied values in solution 1.
thank you..



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 13047
Location: United States (CA)

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
26 Mar 2017, 19:51
Hi All, One of the great 'design aspects' of most GMAT questions is that they can be approached in a variety of ways. As such, just because you got a question correct doesn't necessarily mean that you solved it in the fastest/easiest way possible (and if you're doing a lot of complex math to get to the solution, then you are likely NOT using the fastest approach). Sometimes the simplest approach to certain questions is just 'brute force' arithmetic  plow through the basic math necessary to prove the solution. Here, we're asked for the number of INTEGER solutions to X  6 > 3X + 6. From the answer choices, it seems clear that there aren't that many possibilities (and it's unlikely that there's an 'unlimited' number of solutions), so we just have to find the 1, 3, 5 or 7 options that 'fit' this inequality. To start, let's TEST a couple of simple values for X: 0 and 1.... IF... X = 0, then we end up with... 6 > 6 ..... 6 > 6.... which is NOT correct  so X=0 is NOT a solution IF... X = 1, then we end up with... 5 > 9 ..... 5 > 9.... which is NOT correct  so X=1 is NOT a solution Increasing the value of X will just make the "right side' absolute value a lot bigger, so there's no point in raising X. Thus, let's decrease it and see what happens.... IF... X = 1, then we end up with... 7 > 3 ..... 7 > 3.... which IS correct  so X=1 IS a solution At this point, how long would it take you to TEST 2, 3, 4, 5 and 6? What happens when you try 6 though? And what about 7? Notice how you would stop working for the same reason why you wouldn't both trying 2, 3, 4, etc... Now, how many total solutions do you have? Final Answer: GMAT assassins aren't born, they're made, Rich
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com
Rich Cohen
CoFounder & GMAT Assassin
Special Offer: Save $75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/
*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****



Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1323
Location: Malaysia

For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
01 May 2017, 22:29
Bunuel wrote: For how many integer values of x, is \(x – 6 > 3x + 6\) ?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite Bunuel Why my solution is incorrect? (1) Take positive \(x – 6=x  6\) \(3x + 6=3x + 6\) \(x6>3x+6\) \(x<6\) (2) Take negative \(x6 = x+6\) \(3x+6 = 3x6\) \(x+6>3x6\) \(x>6\) Does this approach correct? https://gmatclub.com/forum/forhowmany ... l#p1804531
_________________
"Be challenged at EVERY MOMENT."
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum  Please DO NOT post short answer in your post!
Advanced Search : https://gmatclub.com/forum/advancedsearch/



Current Student
Joined: 12 Aug 2015
Posts: 2630

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
01 May 2017, 22:55
ziyuen wrote: Bunuel wrote: For how many integer values of x, is \(x – 6 > 3x + 6\) ?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite Bunuel Why my solution is incorrect? (1) Take positive \(x – 6=x  6\) \(3x + 6=3x + 6\) \(x6>3x+6\) \(x<6\) (2) Take negative \(x6 = x+6\) \(3x+6 = 3x6\) \(x+6>3x6\) \(x>6\) Does this approach correct? https://gmatclub.com/forum/forhowmany ... l#p1804531Hi ziyuenIt is not that easy. This is an extremely difficult problem. You highlighted part is incorrect.
Here is what did =>
Firstly the definition of x > x= x for x>0 = x for x<0 =0 for x=0
Now lets look at the two modulus equations > x6 = x6 for x>6 =(x6)=x+6 for x<6 =0 for x=6
Similarly 3x+6 => 3x+6 for x>2 => 3x6 for x<2 => 0 for x=2
Hence the three boundaries are => (∞,2] , [2,6]and [6,∞)
For x>6 ==>
x6>3x+6 x<6
REJECTED as x cannot be x>6 and x<6 simultaneously. ((Also x≠6 as modulus is always positive))
For x=> [2,6) x+6>3x+6 4x4x<0 x<0
Hence x can be 2 or 1
Finally when x<2 ==> x+6 >3x6 2x>12 x>6
Hence x can be 5,4,3
Final conclusion > x can be 5,4,3,2,1
FIVE VALUES.
SMASH THAT C.
P.S> It took me about 3 minutes and 32 seconds to get to the correct answer. This question is extremely helpful to clear basics.
Also you can go through chetan2u 's blog here > http://gmatclub.com/forum/absolutemodu ... l#p1622372 That would surely help you out to clear the Modulus basics.
Regards Stone Cold
_________________
MBA Financing: INDIAN PUBLIC BANKS vs PRODIGY FINANCE! Getting into HOLLYWOOD with an MBA! The MOST AFFORDABLE MBA programs!STONECOLD's BRUTAL Mock Tests for GMATQuant(700+)AVERAGE GRE Scores At The Top Business Schools!



Manager
Joined: 15 Aug 2015
Posts: 56
Location: India
GPA: 3.21

For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
27 Dec 2017, 07:50
Bunuel wrote: For how many integer values of x, is x – 6 > 3x + 6?
(A) 1
(B) 3
(C) 5
(D) 7
(E) Infinite MY way simple and sober and within a minute you get the answerequate x – 6 = 3x + 6we get x=6 Now if you will decrease x let's say Quote: 7 ((try it) then x – 6 > 3x + 6 wil not hold ((LHS WILL BE LESS THAN RHSHowever if you DECREASE X Quote: say 5 then x – 6 > 3x + 6 holds this continues till 1 and then again fails therefore 5.4,3,2,1 check for 0 Ans =5



Intern
Joined: 15 Sep 2017
Posts: 23
GPA: 3.5

Re: For how many integer values of x, is x – 6 > 3x + 6?
[#permalink]
Show Tags
24 Feb 2018, 10:30
x6 > 3x + 6? (X  6)^2 > (3x + 6)^2 x^2  12x + 36 > 9x^2 + 36x + 36 8x^2 + 48x < 0 8x ( x + 6) < 0 Now, for it to be negative, either 1) 8x is positive and (x+6) is negative or 2) 8x is negative and (x+6) is positive. Both cannot be positive and both cannot be negative 1) For 8x to be positive and x+6 to be negative is impossible. As you cannot have a positive number x that when multiplied by 8 is positive and when added to 6 is negative, then no results here. 2) For 8x is negative and (x+6) is positive: Any negative number multiplied by 8 is negative, but not any negative number added to 6 is positive. The only negative integers that when added to 6 become positive are 1, 2, 3, 4 and 5 Therefore, answer C, 5 integers
_________________
"Revenge is a dish best served cold"




Re: For how many integer values of x, is x – 6 > 3x + 6? &nbs
[#permalink]
24 Feb 2018, 10:30






