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Senior Manager  G
Joined: 24 Apr 2016
Posts: 317
If Y = |X + 1| - |X-2|, then  [#permalink]

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25 00:00

Difficulty:   45% (medium)

Question Stats: 65% (02:05) correct 35% (02:02) wrong based on 412 sessions

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If $$Y = |X + 1| - |X-2|$$, then

(a) $$-3 \leq Y \leq 0$$
(b) $$-3 \leq Y \leq 3$$
(c) $$Y \leq -3$$
(d) $$Y \geq -3$$
(e) No Solution
Target Test Prep Representative G
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2812
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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3
6
quantumliner wrote:
If Y = |X + 1| - |X-2|, then

(a) -3 <= Y <=0
(b) -3 <= Y <=3
(c) Y <=-3
(d) Y >=-3
(e) No Solution

Since the expressions in the absolute value will equal zero when X = -1 and X = 2, we need to investigate the following three cases:

i) X < -1, ii) -1 < X < 2, and iii) X > 2

Now let’s analyze each case.

i) If X < -1, we see that both of the expressions inside the absolute values are negative, and thus:

Y = -(X - 1) - [-(X - 2)]

Y = - X -1 + X - 2

Y = -3

ii) If -1 < X < 2, then the first expression is positive but the second expression is negative; thus:

Y = X + 1 - [-(X - 2)]

Y = X + 1 + X - 2

Y = 2X -1

Now, recall that -1 < X < 2, so

-2 < 2X < 4

-3 < 2X - 1 < 3

Since Y = 2X - 1, -3 < Y < 3.

iii) If X > 2, then both expressions inside the absolute value are positive; thus:

Y = X + 1 - (X - 2)

Y = X + 1 - X + 2

Y = 3

Combining the results of the three cases, we see that -3 ≤ Y ≤ 3.

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##### General Discussion
Director  D
Joined: 05 Mar 2015
Posts: 985
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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3
maxschmid wrote:
rohit8865 wrote:

3 ranges to define

1) x< -1 then y= -3
2) -1<=x<2 then y = 2x-1
3) x>= 2 then y=3

as second condition dependent on value of x
thus from range 1 & 2
-3 <= Y <=3

Ans B

How did you came up with the 3 ranges and how to define them? I dont understand what you are doing.
Thanks

maxschmid

for such questions always do the below suggested

Ix+1I will be 0 when x= -1
similarly Ix-2I will be 0 when x=2

thus we have boundaries defined as -1 and 2
so i have checked on both sides of boundaries defined in three steps for value y

1) x< -1 then y= -3
2) -1<=x<2 then y = 2x-1
3) x>= 2 then y=3

hope now u can understand
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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3
1
Hi All,

This question can be solved by TESTing VALUES.

We're told that Y = |X + 1| - |X-2|. We're asked to find the range of value for Y.

IF....
X = 0, then Y = |1| - |-2| = -1, so Y can equal -1. Eliminate Answers C and E. Considering the three remaining answers, we know that Y either has an "upper limit" or it does not, so let's see what happens if we make X really big or really small....

IF...
X = 100, then Y = |101| - |98| = 3. This is interesting, since we've appeared to now randomly hit the 'upper limit' in Answer B. Eliminate Answer A. What if we try something even bigger....
X = 1000, then Y = |1001| - |998| = 3. This is the exact SAME value... It certainly looks like we've found the upper limit, but just to be sure, I'll do one more Test...

IF....
X = -50, then Y = |-49| - |-52| = -3. We clearly have the range now.

GMAT assassins aren't born, they're made,
Rich
_________________

Originally posted by EMPOWERgmatRichC on 16 Feb 2017, 14:13.
Last edited by EMPOWERgmatRichC on 24 Sep 2017, 13:27, edited 1 time in total.
Senior SC Moderator V
Joined: 14 Nov 2016
Posts: 1347
Location: Malaysia
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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2
warriorguy wrote:

I have a query regarding the range. Shouldn't it be -1 <= X < 2 and X >= 2

Dear warriorguy, Graphical illustration will clear your doubt.

When $$x = 2$$

$$Y = |x+1| - |x-2| = |2+1| - |2-2| = |3| + |0| = 3$$

Coordinate 1 $$(2, 3)$$

When $$x = -1$$

$$Y = |x+1| - |x-2| = |-1+1| - |-1-2| = |0| - |3| = -3$$

Coordinate 2 $$(-1, -3)$$

Therefore, the solution is $$-3 ≤ y ≤ 3$$
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Director  D
Joined: 05 Mar 2015
Posts: 985
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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1
2
quantumliner wrote:
If Y = |X + 1| - |X-2|, then

(a) -3 <= Y <=0
(b) -3 <= Y <=3
(c) Y <=-3
(d) Y >=-3
(e) No Solution

3 ranges to define

1) x< -1 then y= -3
2) -1<=x<2 then y = 2x-1
3) x>= 2 then y=3

as second condition dependent on value of x
thus from range 1 & 2
-3 <= Y <=3

Ans B
Intern  B
Joined: 16 Jan 2017
Posts: 5
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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1
rohit8865 wrote:

3 ranges to define

1) x< -1 then y= -3
2) -1<=x<2 then y = 2x-1
3) x>= 2 then y=3

as second condition dependent on value of x
thus from range 1 & 2
-3 <= Y <=3

Ans B

How did you came up with the 3 ranges and how to define them? I dont understand what you are doing.
Thanks
Manager  S
Joined: 30 May 2012
Posts: 197
Location: United States (TX)
Concentration: Finance, Marketing
GPA: 3.3
WE: Information Technology (Consulting)
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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1
EMPOWERgmatRichC wrote:
...

IF....
X = 0, then Y = |1| - |-2| = 1, so Y can equal 1.

...
Rich

How so? How does one approach this via a number line?
Retired Moderator P
Joined: 04 Aug 2016
Posts: 475
Location: India
GPA: 4
WE: Engineering (Telecommunications)
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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1
JeffTargetTestPrep wrote:
quantumliner wrote:
If Y = |X + 1| - |X-2|, then

(a) -3 <= Y <=0
(b) -3 <= Y <=3
(c) Y <=-3
(d) Y >=-3
(e) No Solution

Since the expressions in the absolute value will equal zero when X = -1 and X = 2, we need to investigate the following three cases:

i) X < -1, ii) -1 < X < 2, and iii) X > 2

Now let’s analyze each case.

i) If X < -1, we see that both of the expressions inside the absolute values are negative, and thus:

Y = -(X - 1) - [-(X - 2)]

Y = - X -1 + X - 2

Y = -3

ii) If -1 < X < 2, then the first expression is positive but the second expression is negative; thus:

Y = X + 1 - [-(X - 2)]

Y = X + 1 + X - 2

Y = 2X -1

Now, recall that -1 < X < 2, so

-2 < 2X < 4

-3 < 2X - 1 < 3

Since Y = 2X - 1, -3 < Y < 3.

iii) If X > 2, then both expressions inside the absolute value are positive; thus:

Y = X + 1 - (X - 2)

Y = X + 1 - X + 2

Y = 3

Combining the results of the three cases, we see that -3 ≤ Y ≤ 3.

I have a query regarding the range. Shouldn't it be -1 <= X < 2 and X >= 2
Manager  S
Joined: 30 May 2012
Posts: 197
Location: United States (TX)
Concentration: Finance, Marketing
GPA: 3.3
WE: Information Technology (Consulting)
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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mikemcgarry - Could you help me with plotting a solution on the number line, please?
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
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Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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Hi Blackbox,

Thanks for catching the error; I've updated my explanation.

GMAT assassins aren't born, they're made,
Rich
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Manager  G
Joined: 11 Feb 2013
Posts: 216
Location: United States (TX)
GMAT 1: 490 Q44 V15 GMAT 2: 690 Q47 V38 GPA: 3.05
WE: Analyst (Commercial Banking)
If Y = |X + 1| - |X-2|, then  [#permalink]

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The maximum/Minimum DIFFERENCE between two ABSOLUTE values is THE ABSOLUTE VALUE of their individual DIFFERENCE .
For example, if two values are a&b.
we know, their individual absolute values are /a/ and /b/ and
DIFFERENCE between a & b= a-b.
The maximum/Minimum DIFFERENCE between two ABSOLUTE values= /(a-b)/

Originally posted by BelalHossain046 on 01 Nov 2019, 20:03.
Last edited by BelalHossain046 on 01 Nov 2019, 20:11, edited 1 time in total.
Manager  G
Joined: 11 Feb 2013
Posts: 216
Location: United States (TX)
GMAT 1: 490 Q44 V15 GMAT 2: 690 Q47 V38 GPA: 3.05
WE: Analyst (Commercial Banking)
Re: If Y = |X + 1| - |X-2|, then  [#permalink]

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Back to the question:
Max and min value of y=
y =|X+1|−|X−2|
[The maximum/Minimum DIFFERENCE between two ABSOLUTE values is THE ABSOLUTE VALUE of their individual DIFFERENCE].
=|(x+1)-(x-2)|
=|3|
=+3 or -3
So maximum value of y =3 and minimum value of y= -3.
In other words, Y must be in the range of -3 and +3. Re: If Y = |X + 1| - |X-2|, then   [#permalink] 01 Nov 2019, 20:10
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