Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 25 Oct 2016, 17:25

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Evaluating Mods/Absolute Values Quicker

Author Message
TAGS:

### Hide Tags

Manager
Joined: 12 Feb 2012
Posts: 136
Followers: 1

Kudos [?]: 48 [0], given: 28

### Show Tags

19 Jun 2013, 19:16
I have an equation that looks like this |x – 3| + |x + 1| + |x| = 10 and I want to find the x that satisfies the condition.

I know that -1,0, and 3 are my pivot points that I have to test.

x<-1
-1<=x<=0
0<x<=3

Knowing these are the three ranges where the function changes values is there a quick way to evaluate each subcomponent and know whether it will be positive or negative within that range? I always get stuck on this point because I have to evaluate every subcomponent and figure out what sign i takes with in that range.
For example, in the x<-1 range is each subcomponent evaluates too:
|x-3|=-(x-3) (because |x-3|=-(x-3) if x-3<0, ie, x<3. Because our range is x<-1, clearly within the range of x<3, we place a negative sign)
|x+1|=-(x+1)
and |x|=-x

Just trying to find a quicker way without checking the inequalities to know if I should place a (-) sign on the subcomponents or not.
Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 630
Followers: 78

Kudos [?]: 1054 [0], given: 136

Re: Evaluating Mods/Absolute Values Quicker [#permalink]

### Show Tags

22 Jun 2013, 08:13
alphabeta1234 wrote:
I have an equation that looks like this |x – 3| + |x + 1| + |x| = 10 and I want to find the x that satisfies the condition.

I know that -1,0, and 3 are my pivot points that I have to test.

x<-1
-1<=x<=0
0<x<=3

Knowing these are the three ranges where the function changes values is there a quick way to evaluate each subcomponent and know whether it will be positive or negative within that range? I always get stuck on this point because I have to evaluate every subcomponent and figure out what sign i takes with in that range.
For example, in the x<-1 range is each subcomponent evaluates too:
|x-3|=-(x-3) (because |x-3|=-(x-3) if x-3<0, ie, x<3. Because our range is x<-1, clearly within the range of x<3, we place a negative sign)
|x+1|=-(x+1)
and |x|=-x

Just trying to find a quicker way without checking the inequalities to know if I should place a (-) sign on the subcomponents or not.

We know that |x-0| is the distance of the point x from the origin. Representing all the pivotal points on the number line, the question finally boils down to this : For What value(s) of x, will the distance of x from the points -1,0 and 3 add upto to give 10 units?
Attachment:

Image.jpg [ 8.44 KiB | Viewed 880 times ]

I. $$0<=x<=3$$ Note that for any x,the expression |x|+|x-3| will always give you a CONSTANT sum of distance as 3 units only, i.e. |x|+|x-3| = 3. Also, assuming x to be farthest in this range i.e. at x= 3, the distance between x = 3 and x = -1 is 4 units. Thus, the sum total of all the distances,i.e.
|x – 3| + |x + 1| + |x| is at most 7 units.Thus, no x for [0,3] will ever add upto 10, for the above mentioned expression.

II. $$-1=<x<=0$$ Note the expression |x|+|x+1| will always give you a CONSTANT sum of distance as 1 unit only , i.e. |x|+|x+1| = 1. Again, just as above, assuming x to be farthest in the range i.e. at x = -1, the distance between x = -1 and x = 3 is 4 units. Thus, the sum total of all the distances,i.e.
|x – 3| + |x + 1| + |x| is at most 5 units.Thus, no x for [-1,0] will ever add upto 10, for the above mentioned expression.

III. For $$x>3$$ Notice that we have already found out that the maximum sum of distance for x = 3 is 7 units.Now, if i take any value of x>3, it would give a value which is more than 7 units;so now, we know that THERE IS one unique value of x for this range, which will give the sum of distance as 10 units.Thus, we need an extra 3 units. To find the exact value of x, note that the extra 3 units will be contributed by each of the 3 points (-1,0 and 3) EQUALLY, i.e. 1 unit from each of the 3 pivotal points. Thus, as x gets a distance of 1 unit from 3, hence x = 3+1 = 4. Thus, x = 4 is one solution.

II.For $$x<-1$$ Notice that we have already found the maximum sum of distance for x = -1 is 5 units.As per the problem, this total should be 10 units. Thus, we need an extra 5 units.Just as above, we know for sure that THERE IS another unique value of x, which will give a sum total of 10 units. To find the exact value of x, note that the extra 5 units will be contributed by each of the 3 points (-1,0 and 3) EQUALLY. Thus, as x is at a distance of $$\frac{5}{3}$$ unit from -1 and also x<-1, hence x = $$\frac{-8}{3}$$.

Hope this helps.
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 6971
Location: Pune, India
Followers: 2030

Kudos [?]: 12761 [0], given: 221

Re: Evaluating Mods/Absolute Values Quicker [#permalink]

### Show Tags

27 Jun 2013, 21:30
alphabeta1234 wrote:
I have an equation that looks like this |x – 3| + |x + 1| + |x| = 10 and I want to find the x that satisfies the condition.

I know that -1,0, and 3 are my pivot points that I have to test.

x<-1
-1<=x<=0
0<x<=3

Knowing these are the three ranges where the function changes values is there a quick way to evaluate each subcomponent and know whether it will be positive or negative within that range? I always get stuck on this point because I have to evaluate every subcomponent and figure out what sign i takes with in that range.
For example, in the x<-1 range is each subcomponent evaluates too:
|x-3|=-(x-3) (because |x-3|=-(x-3) if x-3<0, ie, x<3. Because our range is x<-1, clearly within the range of x<3, we place a negative sign)
|x+1|=-(x+1)
and |x|=-x

Just trying to find a quicker way without checking the inequalities to know if I should place a (-) sign on the subcomponents or not.

This question is from my post given here: http://www.veritasprep.com/blog/2011/01 ... s-part-ii/
I have used a graphical approach here which is much faster. It will take some time to wrap your head around it initially but once you do, these questions become very simple.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Re: Evaluating Mods/Absolute Values Quicker   [#permalink] 27 Jun 2013, 21:30
Similar topics Replies Last post
Similar
Topics:
1 absolute values 2 10 Nov 2013, 00:47
what is the value? 1 19 Dec 2010, 08:17
value of probability 1 10 Jun 2010, 13:08
7 What inequality represents the condition 1<x<9? 11 24 May 2010, 18:21
absolute value 2 13 Jan 2010, 10:11
Display posts from previous: Sort by