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Math Expert V
Joined: 02 Sep 2009
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The Reason Behind Absolute Value Questions on the GMAT  [#permalink]

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 FROM Veritas Prep Blog: The Reason Behind Absolute Value Questions on the GMAT Even after working extensively on absolute value questions, sometimes students come up with “why?” i.e. why do we have to take positive and negative values? Why do we have to consider ranges etc. They know the process but they do not understand the reason they need to follow the process. So here today, in this post, we will try to explain the reason.You know how to solve an equation such as x + 2x = 4. Simple enough, right? Just add x with 2x to get 3x and separate out the x on one side. But what do you do when you have an equation with absolute values? How will you handle that equation? Say, you have |x| + 2x = 4. Is this your regular equation? No! You CANNOT say that x + 2x = 4 => 3x = 4 => x = 4/3. You have an absolute value and that complicates matters. You need to get rid of it to get a solution for x. How do you get rid of absolute values? The definition of absolute value helps us here:|x| = x if x >= 0|x| = -x if x < 0So you can substitute x for |x| to make it a regular equation but only if x is non negative. If x is negative, then you put -x instead of |x| to convert it into a simple equation. And that is the reason you need to take positive and negative values of what is inside the absolute value sign.Similarly,|x-5| = (x-5) if (x-5) >= 0 i.e. if x >= 5|x-5| = -(x-5) if (x-5) < 0 i.e. if x < 5Let’s go back to the previous example and see how we can get rid of the absolute value to make it a regular equation:Question 1: What is the value of x given |x| + 2x = 4?We don’t know whether x is positive or negative so we will look at what happens in both cases:Case 1: x is positive or 0If x >= 0 then equation becomes x + 2x = 4 => x = 4/3Our initial condition is that x is non negative. We get a positive solution on solving it and hence 4/3 is a valid solution.Case 2: x is negativeIf x < 0 then equation becomes -x + 2x = 4 => x = 4Our initial condition is that x is negative. We get a positive solution on solving it and hence x = 4 is not a valid solution. Had we obtained a negative solution, it would have been valid.So there is only one solution x = 4/3.We hope the entire process makes more sense now. Let’s follow it up with a complex question from our algebra book.Question 2:  If x and y are integers and y = |x+3| + |4-x|, does y equal 7?Statement 1: x < 4Statement 2: x > -3Solution: Now what do you do when you have y = |x+3| + |4-x|? How do you convert this into a regular equation? You don’t know whether whatever is in the absolute value sign is positive or negative. How will you get rid of the sign then? You will work on all the cases (messy algebra coming up!).Now, we see the same logic in this question:y = |x+3| + |4-x||x+3| = (x+3) if (x+3) >= 0. In other words, if x >= -3|x+3| = -(x+3) if (x+3) < 0. In other words, if x < -3|4-x| = (4-x) if (4-x) >= 0. In other words, if x <= 4|4-x| = -(4-x) if (4-x) < 0. In other words, if x > 4So our absolute values behave differently when x < -3, between -3 and 4 and when x > 4. We say that -3 and 4 are our transition points.Case 1:When x < -3, |x+3| = -(x+3) and |4-x| = (4-x).So the equation becomes y = -(x+3) + (4-x)y = 1 – 2xFor different values of x, y will take different values. Recall that x must be less than -3. Say x = -4, then y = 9. If x = -5, y = 11.Case 2:When -3 <= x <= 4, |x+3| = (x+3) and |4-x| = (4-x).So the equation becomes y = (x+3) + (4-x)y = 7In this range, y will always be 7.Case 3:When x > 4, |x+3| = (x+3) and |4-x| = -(4-x)So the equation becomes y = (x+3) – (4-x)y = 2x – 1For different values of x, y will take different values. Recall that x must be more than 4. Say x = 5, then y is 9. If x = 6, then y is 11.Note that y equals 7 only when x is between -3 and 4. Both statements together tell us that x is between -3 and 4. No statement alone gives us this information. Hence, using both statements, we know that y must be 7.Answer (C)Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!
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IIMA, IIMC School Moderator V
Joined: 04 Sep 2016
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Re: The Reason Behind Absolute Value Questions on the GMAT  [#permalink]

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hi Bunuel

do we have sample OG Qs to master this concept?

WR,
Arpit
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Math Expert V
Joined: 02 Sep 2009
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Re: The Reason Behind Absolute Value Questions on the GMAT  [#permalink]

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hi Bunuel

do we have sample OG Qs to master this concept?

WR,
Arpit

You can mar OG and Absolute values on the search page below to get that:
https://gmatclub.com/forum/search.php?view=search_tags
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Re: The Reason Behind Absolute Value Questions on the GMAT  [#permalink]

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Bunuel wrote:
[rss2posts title="Veritas Prep Blog" title_url="http://www.veritasprep.com/blog/2014/06/the-reason-behind-absolute-value-questions-on-the-gmat/" sub_title="The Reason Behind Absolute Value Questions on the GMAT"] |x| = x if x >= 0

|x| = -x if x < 0

So you can substitute x for |x| to make it a regular equation but only if x is non negative. If x is negative, then you put -x instead of |x| to convert it into a simple equation. And that is the reason you need to take positive and negative values of what is inside the absolute value sign.

I still don't understand WHY |x| = -x when x<0.
To my understanding |x| is the distance of x from zero and distances are not negative. So, then why?
ISB School Moderator P
Joined: 08 Dec 2013
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GMAT 1: 630 Q47 V30 WE: Operations (Non-Profit and Government)
The Reason Behind Absolute Value Questions on the GMAT  [#permalink]

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Karmesh wrote:
Bunuel wrote:
[rss2posts title="Veritas Prep Blog" title_url="http://www.veritasprep.com/blog/2014/06/the-reason-behind-absolute-value-questions-on-the-gmat/" sub_title="The Reason Behind Absolute Value Questions on the GMAT"] |x| = x if x >= 0

|x| = -x if x < 0

So you can substitute x for |x| to make it a regular equation but only if x is non negative. If x is negative, then you put -x instead of |x| to convert it into a simple equation. And that is the reason you need to take positive and negative values of what is inside the absolute value sign.

I still don't understand WHY |x| = -x when x<0.
To my understanding |x| is the distance of x from zero and distances are not negative. So, then why?

What is l-7l, it is the distance of -7, on the number line, from zero.
No let's follow the formula lxl = -(x) iff x<0,
Here x=-7, cool.
so answer is - (-7), i.e. 7.

So, the absolute value (read distance from zero on the number line) of -7 is 7

Hope this helps. Karmesh Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10023
Location: Pune, India
Re: The Reason Behind Absolute Value Questions on the GMAT  [#permalink]

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Karmesh wrote:
Bunuel wrote:
[rss2posts title="Veritas Prep Blog" title_url="http://www.veritasprep.com/blog/2014/06/the-reason-behind-absolute-value-questions-on-the-gmat/" sub_title="The Reason Behind Absolute Value Questions on the GMAT"] |x| = x if x >= 0

|x| = -x if x < 0

So you can substitute x for |x| to make it a regular equation but only if x is non negative. If x is negative, then you put -x instead of |x| to convert it into a simple equation. And that is the reason you need to take positive and negative values of what is inside the absolute value sign.

I still don't understand WHY |x| = -x when x<0.
To my understanding |x| is the distance of x from zero and distances are not negative. So, then why?

You are ignoring the second part of the statement: "when x < 0"
When x itself is NEGATIVE, -x is POSITIVE. Of course, distance is positive. Hence when x itself is positive, |x| is x but when x itself is negative, you need to make it positive by making it -x.
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Karishma
Veritas Prep GMAT Instructor Re: The Reason Behind Absolute Value Questions on the GMAT   [#permalink] 27 Jun 2019, 22:56
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