Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 25 Dec 2009
Posts: 96

What is the solution set for 3x2<=2x5 [#permalink]
Show Tags
16 Jan 2010, 07:00
2
This post received KUDOS
33
This post was BOOKMARKED
Question Stats:
82% (02:58) correct 18% (07:01) wrong based on 89 sessions
HideShow timer Statistics
What is the solution set for \(3x2\leq2x5\)
One way to solve is to square both the terms of course , but what is other way of solving it.



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
16 Jan 2010, 07:34
12
This post received KUDOS
Expert's post
12
This post was BOOKMARKED
GMATMadeeasy wrote: What is the solution set for \(3x2\leq2x5\)
One way to solve is to square both the terms of course , but what is other way of solving it. First you should determine the check points (key points): \(\frac{2}{3}\) and \(\frac{5}{2}\). Hence we'll have three ranges to check: A. \(x<\frac{2}{3}\) > \(3x+2\leq2x+5\) > \(3\leq{x}\), as \(x<\frac{2}{3}\), then \(3\leq{x}<\frac{2}{3}\); B. \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) > \(3x2\leq2x+5\) > \(x\leq\frac{7}{5}\), as \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) , then \(\frac{2}{3}\leq{x}\leq\frac{7}{5}\); C. \(x>\frac{5}{2}\) > \(3x2\leq2x5\) > \(x\leq{3}\), as \(x>\frac{5}{2}\), then in this range we have no solution; Ranges from A and B give us the solution as: \(3\leq{x}\leq\frac{7}{5}\).
_________________
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



Manager
Joined: 25 Dec 2009
Posts: 96

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
16 Jan 2010, 09:15
Thank you , it helps greatly.
Question : What was normal way of doing it back in school? I am wondering how I used to solve them ?
Working out questions from your post and other guys notes on the site in the mean time.
Thanks a lot Bunuel.



Retired Moderator
Status: The last round
Joined: 18 Jun 2009
Posts: 1267
Concentration: Strategy, General Management

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
16 Jan 2010, 10:49
1
This post received KUDOS
@ Bunuel: I didn't pick your strategy. My way of doing is simple but it is giving different answer. 3x2 <= 2x5 We will consider two scenarios 3x2 <= 2x5 & (3x2) <= 2x5 x<= 3 & x >= 7/5 So the range will be x<= 3 & x >= 7/5 Kindly let me know where did I go wrong???
_________________
[ From 470 to 680My Story ] [ My Last Month Before Test ] [ GMAT Prep Analysis Tool ] [ US. Business School Dashboard ] [ Int. Business School Dashboard ]
I Can, I Will
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
16 Jan 2010, 12:04
Hussain15 wrote: @ Bunuel:
I didn't pick your strategy.
My way of doing is simple but it is giving different answer.
3x2 <= 2x5 We will consider two scenarios 3x2 <= 2x5 & (3x2) <= 2x5 x<= 3 & x >= 7/5
So the range will be x<= 3 & x >= 7/5
Kindly let me know where did I go wrong??? If you plug the numbers from the ranges you got, you'll see that the inequality doesn't hold true. As for the solution: we have two absolute values \(3x2\) and \(2x5\). \(3x2\) changes sign at \(\frac{2}{3}\) and \(2x5\) changes sign at \(\frac{5}{2}\). \((I)\frac{2}{3}(II)\frac{5}{2}(III)\) We got three ranges as above. We should expand given inequality in these ranges and see what we'll get. Hope it's clear.
_________________
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



Retired Moderator
Status: The last round
Joined: 18 Jun 2009
Posts: 1267
Concentration: Strategy, General Management

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
16 Jan 2010, 12:39
Thanks for the reply Bunue! But there is one more issue. In range A, you have changed the signs of both modules, in range B you have done it only for 2x+5 and in third case you haven't changed the signs. What's the logic behind this one??
_________________
[ From 470 to 680My Story ] [ My Last Month Before Test ] [ GMAT Prep Analysis Tool ] [ US. Business School Dashboard ] [ Int. Business School Dashboard ]
I Can, I Will
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
16 Jan 2010, 12:54
6
This post received KUDOS
Expert's post
3
This post was BOOKMARKED
Hussain15 wrote: Thanks for the reply Bunue!
But there is one more issue.
In range A, you have changed the signs of both modules, in range B you have done it only for 2x+5 and in third case you haven't changed the signs. What's the logic behind this one?? I got your point. I'm not "changing" the signs, I'm expanding the absolute values in each range. In range A, when \(x<\frac{2}{3}\): \(3x2=3x+2\) and \(2x5=2x+5\), so we get \(3x+2\leq2x+5\). In range B, when \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x+5\), so we get \(3x2\leq2x+5\). In range C, when \(x>\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x5\), so we get \(3x2\leq2x5\).
_________________
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



Manager
Joined: 20 Apr 2010
Posts: 193
Schools: ISB, HEC, Said

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
18 Sep 2010, 07:08
Thanks Bunuel you posted excellent approach to deal with inequalities. I think you should write somthing on inequalities as well. Kudos to you



Intern
Joined: 01 Aug 2010
Posts: 6

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
11 Mar 2011, 11:22
Bunuel wrote: GMATMadeeasy wrote: What is the solution set for \(3x2\leq2x5\)
One way to solve is to square both the terms of course , but what is other way of solving it. First you should determine the check points (key points): \(\frac{2}{3}\) and \(\frac{5}{2}\). Hence we'll have three ranges to check: A. \(x<\frac{2}{3}\) > \(3x+2\leq2x+5\) > \(3\leq{x}\), as \(x<\frac{2}{3}\), then \(3\leq{x}<\frac{2}{3}\); B. \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) > \(3x2\leq2x+5\) > \(x\leq\frac{7}{5}\), as \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) , then \(\frac{2}{3}\leq{x}\leq\frac{7}{5}\); C. \(x>\frac{5}{2}\) > \(3x2\leq2x5\) > \(x\leq{3}\), as \(x>\frac{5}{7}\), then in this range we have no solution; Ranges from A and B give us the solution as: \(3\leq{x}\leq\frac{7}{5}\). that's a lovely and uncomplicated way....



Math Forum Moderator
Joined: 20 Dec 2010
Posts: 1935

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
11 Mar 2011, 13:42
1
This post received KUDOS
Bunuel wrote: Hussain15 wrote: Thanks for the reply Bunue!
But there is one more issue.
In range A, you have changed the signs of both modules, in range B you have done it only for 2x+5 and in third case you haven't changed the signs. What's the logic behind this one?? I got your point. I'm not "changing" the signs, I'm expanding the absolute values in each range. In range A, when \(x<\frac{2}{3}\): \(3x2=3x+2\) and \(2x5=2x+5\), so we get \(3x+2\leq2x+5\). In range B, when \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x+5\), so we get \(3x2\leq2x+5\). In range C, when \(x>\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x5\), so we get \(3x2\leq2x5\). My slow brain is just not getting it!!! How both LHS and RHS get negated for the range x<2/3 and only the RHS gets negated for the range B and nothing for range C. What if there were 4 or 5 ranges; what is the deciding factor as to what gets negated. Could you please help me understand?
_________________
~fluke
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
11 Mar 2011, 13:51
fluke wrote: Bunuel wrote: Hussain15 wrote: Thanks for the reply Bunue!
But there is one more issue.
In range A, you have changed the signs of both modules, in range B you have done it only for 2x+5 and in third case you haven't changed the signs. What's the logic behind this one?? I got your point. I'm not "changing" the signs, I'm expanding the absolute values in each range. In range A, when \(x<\frac{2}{3}\): \(3x2=3x+2\) and \(2x5=2x+5\), so we get \(3x+2\leq2x+5\). In range B, when \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x+5\), so we get \(3x2\leq2x+5\). In range C, when \(x>\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x5\), so we get \(3x2\leq2x5\). My slow brain is just not getting it!!! How both LHS and RHS get negated for the range x<2/3 and only the RHS gets negated for the range B and nothing for range C. What if there were 4 or 5 ranges; what is the deciding factor as to what gets negated. Could you please help me understand? Absolute value properties:When \(x\leq{0}\) then \(x=x\), or more generally when \(some \ expression\leq{0}\) then \(some \ expression\leq{(some \ expression)}\). For example: \(5=5=(5)\); When \(x\geq{0}\) then \(x=x\), or more generally when \(some \ expression\geq{0}\) then \(some \ expression\leq{some \ expression}\). For example: \(5=5\); In range A, when \(x<\frac{2}{3}\): \(3x2<0\) so \(3x2=(3x2)\) and \(2x5<0\) so \(2x5=(2x5)\), and we get \(3x+2\leq2x+5\). In range B, when \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\): \(3x2>0\) so \(3x2=3x2\) and \(2x5<0\) so \(2x5=(2x5)\), so we get \(3x2\leq2x+5\). For more check: mathabsolutevaluemodulus86462.htmlHope it's clear.
_________________
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



Manager
Joined: 19 Dec 2010
Posts: 130

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
19 Mar 2011, 13:44
Great solution bunuel, kudos!



Intern
Joined: 24 Feb 2011
Posts: 4

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
02 Sep 2012, 08:33
Bunuel, when you say as 3<= X and as x< \(\frac{2}{3}\) then 3<=x<=\(\frac{2}{3}\) does this mean that the range of x that satisfies the condition x<\(\frac{2}{3}\) is all those points on the number line which satisfy both inequalities.
Also you say ranges from A and B give us solution as 3<= x<= \(\frac{7}{5}\) . Can you please explain this to me. Does this mean that the final solution to this problem would be all those ranges that have a solution to their respective conditions i.e the conditions x<\(\frac{2}{3}\) , \(\frac{2}{3}\)<=x<\(\frac{5}{2}\) and x>=\(\frac{5}{2}\).
Also if for example ranges from A and B were 5<=x<2 and 4<=x<10, then would the final solution be both these ranges or do they have to have an overlap? I know this is a supremely dumb and elementary question, but unfortunately inequalities and modulus happen to be my weakest areas. I cannot express in words my gratitude to you for having put up the GMAT Math Book. That thing is my Math Bible. You've made math so easy to understand.
Best wishes and many many thanks !!! Amogh.



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
03 Sep 2012, 05:04
Amogh wrote: Bunuel, when you say as 3<= X and as x< \(\frac{2}{3}\) then 3<=x<=\(\frac{2}{3}\) does this mean that the range of x that satisfies the condition x<\(\frac{2}{3}\) is all those points on the number line which satisfy both inequalities.
Also you say ranges from A and B give us solution as 3<= x<= \(\frac{7}{5}\) . Can you please explain this to me. Does this mean that the final solution to this problem would be all those ranges that have a solution to their respective conditions i.e the conditions x<\(\frac{2}{3}\) , \(\frac{2}{3}\)<=x<\(\frac{5}{2}\) and x>=\(\frac{5}{2}\).
Also if for example ranges from A and B were 5<=x<2 and 4<=x<10, then would the final solution be both these ranges or do they have to have an overlap? I know this is a supremely dumb and elementary question, but unfortunately inequalities and modulus happen to be my weakest areas. I cannot express in words my gratitude to you for having put up the GMAT Math Book. That thing is my Math Bible. You've made math so easy to understand.
Best wishes and many many thanks !!! Amogh. We consider three ranges: A. \(x<\frac{2}{3}\); B. \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\); C. \(x>\frac{5}{2}\). In range A we get: \(3\leq{x}<\frac{2}{3}\); In range B we get: \(\frac{2}{3}\leq{x}\leq\frac{7}{5}\); In range C there is no solution. So, the given inequality holds true for \(3\leq{x}<\frac{2}{3}\) and \(\frac{2}{3}\leq{x}\leq\frac{7}{5}\). Now, we can combine these two ranges and write: \(3\leq{x}\leq\frac{7}{5}\). Next, the solution set is \(3\leq{x}\leq\frac{7}{5}\), means that any x from \(3\leq{x}\leq\frac{7}{5}\) will satisfy \(3x2\leq2x5\). If for some question we had 5<=x<2 from one range and 4<=x<10 from another, then the solution will be both ranges (no need for overlap). Hope it's clear.
_________________
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



Intern
Joined: 11 Jul 2012
Posts: 42

Re: What is the solution set for 3x2<=2x5 [#permalink]
Show Tags
01 Nov 2012, 12:54
Hello Bunuel. Does this approach always work?: 3x2 <= 2x5 ===> (3x2)^2<= (2x5)^2 ===> 5x^2+ 8x 21 <= 0 solution of the ineq (3, 7/5) Brother Karamazov



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: What is the solution set for 3x2<=2x5 [#permalink]
Show Tags
01 Nov 2012, 13:32
Ousmane wrote: Hello Bunuel. Does this approach always work?: 3x2 <= 2x5 ===> (3x2)^2<= (2x5)^2 ===> 5x^2+ 8x 21 <= 0 solution of the ineq (3, 7/5) Brother Karamazov Do you mean squaring? If both parts of an inequality are nonnegative (as in our case), then you can safely raise both parts to an even power (for example square). A. 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).For example: \(2<4\) > we can square both sides and write: \(2^2<4^2\); \(0\leq{x}<{y}\) > we can square both sides and write: \(x^2<y^2\); But if either of side is negative then raising to even power doesn't always work. For example: \(1>2\) if we square we'll get \(1>4\) which is not right. So if given that \(x>y\) then we can not square both sides and write \(x^2>y^2\) if we are not certain that both \(x\) and \(y\) are nonnegative. B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).For example: \(2<1\) > we can raise both sides to third power and write: \(2^3=8<1=1^3\) or \(5<1\) > \(5^2=125<1=1^3\); \(x<y\) > we can raise both sides to third power and write: \(x^3<y^3\). Hope 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



VP
Status: Final Lap Up!!!
Affiliations: NYK Line
Joined: 21 Sep 2012
Posts: 1066
Location: India
GMAT 1: 410 Q35 V11 GMAT 2: 530 Q44 V20 GMAT 3: 630 Q45 V31
GPA: 3.84
WE: Engineering (Transportation)

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
02 Nov 2012, 05:58
1
This post received KUDOS
Bunuel wrote: GMATMadeeasy wrote: What is the solution set for \(3x2\leq2x5\)
One way to solve is to square both the terms of course , but what is other way of solving it. First you should determine the check points (key points): \(\frac{2}{3}\) and \(\frac{5}{2}\). Hence we'll have three ranges to check: A. \(x<\frac{2}{3}\) > \(3x+2\leq2x+5\) > \(3\leq{x}\), as \(x<\frac{2}{3}\), then \(3\leq{x}<\frac{2}{3}\); B. \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) > \(3x2\leq2x+5\) > \(x\leq\frac{7}{5}\), as \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) , then \(\frac{2}{3}\leq{x}\leq\frac{7}{5}\); C. \(x>\frac{5}{2}\) > \(3x2\leq2x5\) > \(x\leq{3}\), as \(x>\frac{5}{7}\), then in this range we have no solution; Ranges from A and B give us the solution as: \(3\leq{x}\leq\frac{7}{5}\). Hi Bunnuel excellent approach My doubt in range c is whether the fraction is 5/2 or 5/7 if it is the latter can you pls explain how. Regards Archit



VP
Status: Final Lap Up!!!
Affiliations: NYK Line
Joined: 21 Sep 2012
Posts: 1066
Location: India
GMAT 1: 410 Q35 V11 GMAT 2: 530 Q44 V20 GMAT 3: 630 Q45 V31
GPA: 3.84
WE: Engineering (Transportation)

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
02 Nov 2012, 06:22
Bunuel wrote: Hussain15 wrote: Thanks for the reply Bunue!
But there is one more issue.
In range A, you have changed the signs of both modules, in range B you have done it only for 2x+5 and in third case you haven't changed the signs. What's the logic behind this one?? I got your point. I'm not "changing" the signs, I'm expanding the absolute values in each range. In range A, when \(x<\frac{2}{3}\): \(3x2=3x+2\) and \(2x5=2x+5\), so we get \(3x+2\leq2x+5\). In range B, when \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x+5\), so we get \(3x2\leq2x+5\). In range C, when \(x>\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x5\), so we get \(3x2\leq2x5\). While solving for A i understand that since (3x2)<0 so you have multiplied the (3x5) with negetive. But (2x5) is >0 so i didnt understand why did you multiply it with negetive. Is it that you plugged a value 1/2 which is less than 2/3 and than analysed that we have to multiply negetive value. Pls explain



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
02 Nov 2012, 08:47
Archit143 wrote: Bunuel wrote: GMATMadeeasy wrote: What is the solution set for \(3x2\leq2x5\)
One way to solve is to square both the terms of course , but what is other way of solving it. First you should determine the check points (key points): \(\frac{2}{3}\) and \(\frac{5}{2}\). Hence we'll have three ranges to check: A. \(x<\frac{2}{3}\) > \(3x+2\leq2x+5\) > \(3\leq{x}\), as \(x<\frac{2}{3}\), then \(3\leq{x}<\frac{2}{3}\); B. \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) > \(3x2\leq2x+5\) > \(x\leq\frac{7}{5}\), as \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\) , then \(\frac{2}{3}\leq{x}\leq\frac{7}{5}\); C. \(x>\frac{5}{2}\) > \(3x2\leq2x5\) > \(x\leq{3}\), as \(x>\frac{5}{7}\), then in this range we have no solution; Ranges from A and B give us the solution as: \(3\leq{x}\leq\frac{7}{5}\). Hi Bunnuel excellent approach My doubt in range c is whether the fraction is 5/2 or 5/7 if it is the latter can you pls explain how. Regards Archit Yes, it should be 5/2. Typo edited. Thank you. +1.
_________________
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



Math Expert
Joined: 02 Sep 2009
Posts: 44336

Re: Inequalities  Challenging and tricky One [#permalink]
Show Tags
02 Nov 2012, 08:50
Archit143 wrote: Bunuel wrote: Hussain15 wrote: Thanks for the reply Bunue!
But there is one more issue.
In range A, you have changed the signs of both modules, in range B you have done it only for 2x+5 and in third case you haven't changed the signs. What's the logic behind this one?? I got your point. I'm not "changing" the signs, I'm expanding the absolute values in each range. In range A, when \(x<\frac{2}{3}\): \(3x2=3x+2\) and \(2x5=2x+5\), so we get \(3x+2\leq2x+5\). In range B, when \(\frac{2}{3}\leq{x}\leq\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x+5\), so we get \(3x2\leq2x+5\). In range C, when \(x>\frac{5}{2}\): \(3x2=3x2\) and \(2x5=2x5\), so we get \(3x2\leq2x5\). While solving for A i understand that since (3x2)<0 so you have multiplied the (3x5) with negetive. But (2x5) is >0 so i didnt understand why did you multiply it with negetive. Is it that you plugged a value 1/2 which is less than 2/3 and than analysed that we have to multiply negetive value. Pls explain For A: if \(x<\frac{2}{3}\) then both 3x2 and 2x5 are negative. Just substitute x=0<2/3 to check. Hope 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




Re: Inequalities  Challenging and tricky One
[#permalink]
02 Nov 2012, 08:50



Go to page
1 2
Next
[ 35 posts ]



