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For everyone interested in modulus practices, I have listed, [#permalink]
 03 Dec 2006, 05:22
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For everyone interested in modulus practices, I have listed, at the bottom of this topic, several threads of modulus in a graduated level of difficulty.
Keep in mind that it's for trainning purposes and that the most advanced posts are not reprensative of the real GMAT.
Finally, enjoy !..... Fig  a) |1/(x-2)| >= 4, b)|x-3|<2x-4 c)x + 2 < |x2 - 4| solve for x I will highly appreciate detailed explanations, thanks
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A. |1/(x-2)| >= 4
As we are working positive value and as the function f(x) = 1/x decreases when x increases on the domain of x > 0, we can imply that:
|x-2| =< 1/4
That means,
o if x > 2 then,
x-2 =< 1/4
<=> x <= 1/4+2
<=> x <= 9/4
o if x < 2 then,
-x+2 =< 1/4
<=> x >= 2 - 1/4
<=> x >= 7/4
Thus, 7/4 <= x <= 9/4
Last edited by Fig on 29 Dec 2006, 13:57, edited 2 times in total.
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B. |x-3|<2x-4
o If x < 3, then the inequation becomes:
-(x-3)<2x-4
<=> 7 < 3x
<=> x > 7/3
=> 7/3 < x < 3
o If x >= 3, then the inequation becomes:
x-3<2x-4
<=> x > 1
=> x >= 3
Finally, we have : x > 7/3
Last edited by Fig on 04 Dec 2006, 13:40, edited 1 time in total.
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C. x + 2 < |x2 - 4|
|x2 - 4| = |(x+2)(x-2)|
Since the sign of a*x^2+b*x+c is that of a for any values of x not between the 2 roots, solutions of the equation a*x^2+b*x+c = 0 if they exist.
Thus,
o Sign((x+2)(x-2)) > 0 when x > 2 or x < -2
o Sign((x+2)(x-2)) < 0 when -2 < x < 2
If -2 < x < 2, the inequation becomes :
x + 2 < -(x^2 - 4)
<=> x^2 + x - 2 < 0 (B)
Delta = 1 + 4*2 = 9.
The roots are:
o Root1 = (-1 - 3)/2 = -2
o Root2 = (-1 + 3)/2 = 1
So, (B) is true when x is between the roots of x^2 + x - 2. That implies -2 < x < 1.
If x < -2 or x > 2, the inequation becomes :
x + 2 < x^2 - 4
<=> x^2 - x - 6 > 0 (A)
Delta = 1 + 4*6 = 25.
The roots are:
o Root1 = (1 - 5)/2 = -2
o Root2 = (1 + 5)/2 = 3
(A) : x^2 - x - 6 > 0
<=> (x+2)(x-3) > 0
One more time, a is positive here. Thus (x+2)(x-3) > 0 outside of the roots.
In other words, x > 3 or x < -2
Finally, the domain of solutions of x is : x > 3 or x < -2 or -2 < x < 1.
Last edited by Fig on 29 Dec 2006, 04:26, edited 2 times in total.
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FIG I HAVE A QUESTION HERE ABOUT THE ANSWER
THE RANGE , 7/4 <= x <= 9/4 INCLUDES 2 , AND THE FUNCTION IS NOT DEFINED AT X=2 AM I RIGHT???
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yezz wrote: FIG I HAVE A QUESTION HERE ABOUT THE ANSWER
THE RANGE , 7/4 <= x <= 9/4 INCLUDES 2 , AND THE FUNCTION IS NOT DEFINED AT X=2 AM I RIGHT???
Yes .... U are right... My conclusion should clarify it .... We have to remove the x=2
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Hello Fig , hope u r fine and in best health ,
FOR THE THIRD PROBLEM , I BELIEVE THE ANSWER MISS THE RANGE
-2<X<1
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yezz wrote: Hello Fig , hope u r fine and in best health ,
FOR THE THIRD PROBLEM , I BELIEVE THE ANSWER MISS THE RANGE
-2<X<1
I'm ok thanks... it follows the way
Great catch .... Yes, I did a "silly" mistake 2-4 = 6 and ended up with a wrong x^2 + x + 6 .... I have corrected my post following your catch... very good :D
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Thanks man , this is no catch this the fruits of your help and coaching 
thanks a lot
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Fig wrote: B. |x-3|<2x-4
o If x < 3, then the inequation becomes:
-(x-3)<2x-4
<=> 7 < 3x
<x> 7/3
=> 7/3 < x <3>= 3, then the inequation becomes:[/b]
x-3<2x-4
<x> 1
=> x >= 3
Finally, we have : x > 7/3
Hi Fig,
Why have you mentioned that x >=3 and not that x > 3.. I see absolute Questions which sometimes have the >= sign and sometimes the > sign only.. Might be a very mindless question  but hope you would answer..
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ajay_gmat wrote: Fig wrote: B. |x-3|<2x-4
o If x < 3, then the inequation becomes:
-(x-3)<2x-4
<=> 7 < 3x
<x> 7/3
=> 7/3 < x <3>= 3, then the inequation becomes:[/b]
x-3<2x-4
<x> 1
=> x >= 3
Finally, we have : x > 7/3 Hi Fig, Why have you mentioned that x >=3 and not that x > 3.. I see absolute Questions which sometimes have the >= sign and sometimes the > sign only.. Might be a very mindless question but hope you would answer.. Not meaningless but rather precised question ...
Actually, it's just to avoid adding 1 solution alone from another interval... and so to meet a higher degree of "perfection" by making it a smoother way to the final solution and a straightforward simplicity to understand
Note that nothing is wrong if we say > or >= as soon as we do not miss to consider all values of x, especially here the ones that limit 2 intervals
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Fig wrote: A. |1/(x-2)| >= 4
As we are working positive value and as the function f(x) = 1/x decreases when x increases on the domain of x > 0, we can imply that:
|x-2| =< 1/4
can someone please explain why this happens. Why can't I just use |1/x-2| >=4. However, when x<2, I get x<=7/4. Also I agree with everyone else as the range include 2, doesn't lie outside of our limitations.
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Manager
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Fig wrote: B. |x-3|<2x-4
o If x < 3, then the inequation becomes:
-(x-3)<2x-4
<=> 7 < 3x
<=> x > 7/3
=> 7/3 < x < 3
o If x >= 3, then the inequation becomes:
x-3<2x-4
<=> x > 1
=> x >= 3
Finally, we have : x > 7/3 Hi buddy could you please explain this, <=> x > 7/3 => 7/3 < x < 3 <=> x > 1 => x >= 3 i solved and got x>1 & x>7/3
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Re: Absolute Mpractices [Modules] - List of GMAT Questions [#permalink]
12 Aug 2010, 05:10
in case If -2 < x < 2, shouldn't it also have one more solution to problem.i.e. x<-2 and x>1
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Fig wrote: C. x + 2 < |x2 - 4|
|x2 - 4| = |(x+2)(x-2)|
Since the sign of a*x^2+b*x+c is that of a for any values of x not between the 2 roots, solutions of the equation a*x^2+b*x+c = 0 if they exist.
Thus,
o Sign((x+2)(x-2)) > 0 when x > 2 or x < -2
o Sign((x+2)(x-2)) < 0 when -2 < x < 2
Trying hard to grasp this concept, but don't think I'm understanding what you're trying to convey here. Ideally, I would try to solve this as such: Given that |x^2 - 4| is same as |(x+2)(x-2)| Solving for x in |x^2 - 4| > x + 2 in the following two scenarios: 1. When (x+2)(x-2) >= 0 ==> x^2 - 4 > x + 2 ==> x>-1 and x>2 2. When (x+2)(x-2) < 0 ==> -(x^2 - 4) > x + 2 ==> 4 - x^2 > x + 2 ==> x<1 and x<-2 Does the combination of these solutions for x represent the final answer? Am I missing a step?
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fortsill wrote: Fig wrote: C. x + 2 < |x2 - 4|
|x2 - 4| = |(x+2)(x-2)|
Since the sign of a*x^2+b*x+c is that of a for any values of x not between the 2 roots, solutions of the equation a*x^2+b*x+c = 0 if they exist.
Thus,
o Sign((x+2)(x-2)) > 0 when x > 2 or x < -2
o Sign((x+2)(x-2)) < 0 when -2 < x < 2
Trying hard to grasp this concept, but don't think I'm understanding what you're trying to convey here. Ideally, I would try to solve this as such: Given that |x^2 - 4| is same as |(x+2)(x-2)| Solving for x in |x^2 - 4| > x + 2 in the following two scenarios: 1. When (x+2)(x-2) >= 0 ==> x^2 - 4 > x + 2 ==> x>-1 and x>2 2. When (x+2)(x-2) < 0 ==> -(x^2 - 4) > x + 2 ==> 4 - x^2 > x + 2 ==> x<1 and x<-2 Does the combination of these solutions for x represent the final answer? Am I missing a step? First of all you won't see such equation on the GMAT, but anyway: |x^2-4|>x+2. We should consider the cases when x^2-4<0 and x^2-4\geq{0}: x^2-4<0 for -2<x<2 --> -(x^2-4)>x+2 --> x^2+x-2<0 --> (x+2)(x-1)<0--> -2<x<1. Since we are considering the range -2<x<2 then the solution for this case will be intersections of these two (common part): -2<x<1; x^2-4\geq{0} for x\leq{-2} or x\geq{2} --> (x^2-4)>x+2 --> x^2-x-6>0 --> (x+2)(x-3)>0--> x<-2 or x>3. Since we are considering the range x\leq{-2} or x\geq{2} then the solution for this case will be intersections of these four (common parts): x<-2 or x>3; Finally the ranges of x for which |x^2-4|>x+2 true are: x<-2, -2<x<1 and x>3. Solving inequalities: x2-4x-94661.html#p731476inequalities-trick-91482.htmldata-suff-inequalities-109078.htmlrange-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535everything-is-less-than-zero-108884.html?hilit=extreme#p868863Hope it helps.
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