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# Math: Absolute value (Modulus)

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Re: Math: Absolute value (Modulus) [#permalink]  17 Jun 2014, 08:16
My questions: if we have 3 points why do we have 4 conditions? Next, for each of the letters (a,b,c,d) how do we know which equation to use and why is the inequality used? For example, for a) we do x = -8, but then how to we know the equation to use is -(x+3) - (4-x) = -(8+x), and how did the negative sign come in front of the -x+3 and why? I have the same type of questions for b,c,d so maybe an answer to a will help
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Re: Math: Absolute value (Modulus) [#permalink]  17 Jun 2014, 08:57
What does xe mean in the Math Book? (ex : abs value (x) < 5 is equal to x e (-5 , 5)
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Re: Math: Absolute value (Modulus) [#permalink]  17 Jun 2014, 09:07
In the absolute value question, I don't get why the distance between A and X is not explained. There's discussion about Y and A, Y and B, X between A and Y, distance between X and B and Y and B but not A and X , I need a reexplanation, maybe with numbers
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Re: Math: Absolute value (Modulus) [#permalink]  17 Jun 2014, 19:27
Expert's post
sagnik242 wrote:
My questions: if we have 3 points why do we have 4 conditions? Next, for each of the letters (a,b,c,d) how do we know which equation to use and why is the inequality used? For example, for a) we do x = -8, but then how to we know the equation to use is -(x+3) - (4-x) = -(8+x), and how did the negative sign come in front of the -x+3 and why? I have the same type of questions for b,c,d so maybe an answer to a will help

Plot the 3 points: -8, -3 and 4 on the number line. Now there are 4 distinct regions on the number line:
The part that lies to the left of -8,
the part that lies between -8 and -3,
the part that lies between -3 and 4 and
the part that lies to the right of 4.

In each of these regions, each of the absolute value expressions will behave differently.

You should know the definition of absolute value:

|x| = x if x >= 0
|x| = -x if x < 0

Consider the equation:
|x+3| - |4-x| = |8+x|

What happens in the region which is at the left of -8 say x = -10
x+3 = -10+3 = -7
Since x+3 is negative, |x+3| = -(x + 3)

4-x = 4 - (-10) = 14
Since 4-x is positive, |4 - x| = 4 - x

8 + x = 8 - 10 = -2
Since x+8 is negative, |8+x| = -(8+x)

This is the logic used. Now repeat it for each region.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 23 Jul 2013 Posts: 9 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Math: Absolute value (Modulus) [#permalink] 31 Jul 2014, 10:21 i am really confused in case of double/triple modulus. Someone please help me! --> how to open modulus in case of double/triple mod. |x+3| - |4-x| = |8+x| Here origins will be -8, -3 and 4 (I am crystal clear till this point) So we have 4 regions ==> x>4, -3<x<4 , -8<x<-3, and x<-8 I do not understand how to open modulus in these 4 regions? (when to consider positive/negative signs while opening modulus) Math Expert Joined: 02 Sep 2009 Posts: 27468 Followers: 4311 Kudos [?]: 42201 [0], given: 5957 Re: Math: Absolute value (Modulus) [#permalink] 31 Jul 2014, 10:25 Expert's post AkshayDavid wrote: i am really confused in case of double/triple modulus. Someone please help me! --> how to open modulus in case of double/triple mod. |x+3| - |4-x| = |8+x| Here origins will be -8, -3 and 4 (I am crystal clear till this point) So we have 4 regions ==> x>4, -3<x<4 , -8<x<-3, and x<-8 I do not understand how to open modulus in these 4 regions? (when to consider positive/negative signs while opening modulus) Please go through the thread. This question is extensively discussed there. _________________ Intern Joined: 12 Feb 2014 Posts: 3 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: Math: Absolute value (Modulus) [#permalink] 09 Aug 2014, 11:04 can someone help me with this, |x-1|+|x-3|+|x-5|>=9, the solution that I came up for this was completely different than in the book I am working with. Can someone please tell how to proceed with this inequality. The solution I got was (-infinity,1)U(5,infinity). TIA Intern Joined: 22 Aug 2014 Posts: 1 Schools: Booth PT '18 Followers: 0 Kudos [?]: 0 [0], given: 19 Re: Math: Absolute value (Modulus) [#permalink] 15 Oct 2014, 22:25 I dont seem to understand that in the last option (d), why the sign of (4-x) is reversed to positive? Math Expert Joined: 02 Sep 2009 Posts: 27468 Followers: 4311 Kudos [?]: 42201 [0], given: 5957 Re: Math: Absolute value (Modulus) [#permalink] 15 Oct 2014, 23:42 Expert's post mina89 wrote: I dont seem to understand that in the last option (d), why the sign of (4-x) is reversed to positive? When $$x \geq 4$$, |4 - x|=-(4 - x), hence -|4 - x| = -(-(4 - x)) = (4 - x) _________________ Intern Joined: 08 Oct 2014 Posts: 11 Followers: 0 Kudos [?]: 1 [0], given: 50 Re: Math: Absolute value (Modulus) [#permalink] 30 Oct 2014, 22:30 In the Example #1 of "Tips & Tricks" Section what is the significance and reason of the following: "Only B and D has 8/2=4 on the right side and D had left site 0 at x=5. Therefore, answer is D." I have not understood this part at all. Thanks Intern Joined: 22 Oct 2014 Posts: 15 Followers: 0 Kudos [?]: 6 [0], given: 71 Re: Math: Absolute value (Modulus) [#permalink] 08 Dec 2014, 08:31 VeritasPrepKarishma wrote: If |x- 5| < 4, now we are looking for points at a distance less than 4 away from 5. Attachment: Ques2.jpg VeritasPrepKarishma wrote: OR consider that lx+3l>3 means distance of x from -3 is more than 3. If you go to 3 steps to right from -3, you reach 0. Anything after than is ok. If you go 3 steps to left from -3, you reach -6. Anything to its left is ok. Attachment: Ques2.jpg I didn't understand the highlighted portion, Could you please explain: 1. How we are getting to positive 5 from -5 inside of the absolute value; then similarly negative 3 from +3 inside of the absolute value? 2. I can work with the more than or less than from the greater or less than sign but how can I get the value next to "from" , inside of the absolute value, to make a number line? TIA! Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5539 Location: Pune, India Followers: 1369 Kudos [?]: 6966 [0], given: 178 Re: Math: Absolute value (Modulus) [#permalink] 08 Dec 2014, 20:09 Expert's post appleid wrote: VeritasPrepKarishma wrote: If |x- 5| < 4, now we are looking for points at a distance less than 4 away from 5. Attachment: Ques2.jpg VeritasPrepKarishma wrote: OR consider that lx+3l>3 means distance of x from -3 is more than 3. If you go to 3 steps to right from -3, you reach 0. Anything after than is ok. If you go 3 steps to left from -3, you reach -6. Anything to its left is ok. Attachment: Ques2.jpg I didn't understand the highlighted portion, Could you please explain: 1. How we are getting to positive 5 from -5 inside of the absolute value; then similarly negative 3 from +3 inside of the absolute value? 2. I can work with the more than or less than from the greater or less than sign but how can I get the value next to "from" , inside of the absolute value, to make a number line? TIA! Here is the post that explains this logic: http://www.veritasprep.com/blog/2011/01 ... edore-did/ It shows you how to deal with absolute values using the number line. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Math: Absolute value (Modulus) [#permalink]  22 Jan 2015, 07:24
Hello,

There are a few things I don't find clear in the theory about absolute value, and particularly the 3 step method.

1) How do we decide which are the points we are interested in?
In this case for example, |x+3| - |4-x|=|8+x|, why are we interested in -8, -3, 4 and not -4? Is it because if we replace -x with 4 the expression becomes zero? So, in this case we are not saying that we need -4, becuase the minus sign is already there in -x?
2) Why are these values ordered like -8, -3, 4? Are they going from the least to the highest?
3) How are we choosing our conditions? So, why are the conditions these: x<-8, -8<x<=-3, -3<=x<=4 and x>=4? Does this have to do with the order of the values, as mentioned in 3 above? Does a negative sign go with a x< and a positive with a x>=?
4) How do we decide how to change the sign while we compute the inequality? E.g, when x<-8, why is it -(x+3) - (4-x) = - (8+x) or when -8<x<=-3 it is -(x+3) - (4-x) = (8+x)?

Thank you.
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Re: Math: Absolute value (Modulus) [#permalink]  29 Jan 2015, 09:32
Expert's post
pacifist85 wrote:
1) How do we decide which are the points we are interested in?
In this case for example, |x+3| - |4-x|=|8+x|, why are we interested in -8, -3, 4 and not -4? Is it because if we replace -x with 4 the expression becomes zero? So, in this case we are not saying that we need -4, becuase the minus sign is already there in -x?
2) Why are these values ordered like -8, -3, 4? Are they going from the least to the highest?
3) How are we choosing our conditions? So, why are the conditions these: x<-8, -8<x<=-3, -3<=x<=4 and x>=4? Does this have to do with the order of the values, as mentioned in 3 above? Does a negative sign go with a x< and a positive with a x>=?
4) How do we decide how to change the sign while we compute the inequality? E.g, when x<-8, why is it -(x+3) - (4-x) = - (8+x) or when -8<x<=-3 it is -(x+3) - (4-x) = (8+x)?

1) When the expression under modulus equals 0. The expression may or may not change its sign in those special points.
2) Yes, because the number line goes from -infinity all the way to +infinity. -4 is not a special point (see #1; 4 - (-4) = 8 ≠ 0)
3) All conditions cover the number line and break at the special points.
4) Using the definition of modulus: for y = |x|, y = x if x >=0 and y = -x if x < 0.
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Math: Absolute value (Modulus) [#permalink]  23 Feb 2015, 00:29
gettinit wrote:
Let’s consider following examples,

a) $$x < -8$$. $$-(x+3) - (4-x) = -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) $$-8 \leq x < -3$$. $$-(x+3) - (4-x) = (8+x)$$ --> $$x = -15$$. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) $$-3 \leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) $$x \geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)

I understand (a) and (d)... but i don't really understand (b) and (c).
Firstly for (b), since x<0, shouldn't the (-)sign be applied to all factors instead of just (x+3) and (4-x)? Why is the (-)sign not applied to (8+x)?
as for (c), x can be >0 or <0, so what should be done here?

Please do explain why this is so.

Thanks.
Math: Absolute value (Modulus)   [#permalink] 23 Feb 2015, 00:29

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