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# If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7?

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If x and y are integers and y = |x + 3| + |4 - x|, does y equal 7? [#permalink]

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29 Jan 2010, 11:55
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If x and y are integers and y = |x + 3| + |4 - x|, does y equal 7?

(1) x < 4
(2) x > -3
[Reveal] Spoiler: OA

Last edited by Bunuel on 27 Jul 2015, 14:57, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equal 7? [#permalink]

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29 Jan 2010, 12:24
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Expert's post
if x and y are integers and y=|x+3| +|4-x|, does y equal 7

If we carefully look at the question we can see that "does y equal 7" actually means that y doesn't depend on x and then you can see that if we open two moduli with same signs (++ or --), x and -x disappear and 3+4 is 7. So, let's see when we can open moduli ++ or --:

++) x>-3 and x<4 or x e (-3,4)
--) x<-3 and x>4 - it can't be.

So, if x between -3 and 4, y =7. Now, look at our statements: it is obvious that we need two statements.

C.
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If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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26 Sep 2010, 21:26
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If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7?

(1) x < 4
(2) x > -3
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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26 Sep 2010, 23:06
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thirst4edu wrote:
If x & y are integers and y=|x+3| + |4-x|, does y equals 7?

1) x < 4
2) x > -3

Had a hard time solving this, would like to know how to solve this using number picking approach as well as algebraic approach. Thanks.

OA is
[Reveal] Spoiler:
C

$$y=|x+3|+|4-x|$$ two check points: $$x=-3$$ and $$x=4$$ (check point: the value of $$x$$ when expression in || equals to zero), hence three ranges to consider:

A. $$x<{-3}$$ --> $$y=| x + 3| +|4-x| =-x-3+4-x=-2x+1$$, which means that when $$x$$ is in the range {-infinity,-3} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range);

B. $$-3\leq{x}\leq{4}$$ --> $$y=|x+3|+|4-x|=x+3+4-x=7$$, which means that when $$x$$ is in the range {-3,4} the value of $$y$$ is $$7$$ (value of y does not depend on value of $$x$$, when $$x$$ is from the given range);

C. $$x>{4}$$ --> $$y=|x+3|+|4-x|=x+3-4+x=2x-1$$, which means that when $$x$$ is in the range {4, +infinity} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range).

Hence we can definitely conclude that $$y=7$$ if $$x$$ is in the range {-3,4}

(1) $$x<4$$ --> not sufficient ($$x<4$$ but we don't know if it's $$\geq{-3}$$);
(2) $$x>-3$$ --> not sufficient ($$x>-3$$ but we don't know if it's $$\leq{4}$$);

(1)+(2) $$-3<x<4$$ exactly the range we needed, so $$y=7$$. Sufficient.

OR: looking at $$y=|x+3|+|4-x|$$ you can notice that $$y=7$$ ($$y$$ doesn't depend on the value of $$x$$) when $$x+3$$ and $$4-x$$ are both positive, in this case $$x-es$$ cancel out each other and we would have $$y=|x+3|+|4-x|=x+3+4-x=7$$. Both $$x+3$$ and $$4-x$$ are positive in the range $$-3<{x}<4$$ ($$x+3>0$$ --> $$x>-3$$ and $$4-x>0$$ --> $$x<4$$).

Hope it's clear.
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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26 Sep 2010, 23:09
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Let's first solve |x+3|+|4-x|=7 to answer "when is this true ?"

You can solve algebraically but it is much easier to do it using a simple number line approach. Remember |x-a| means distance between x and a on the number line
Here the two points in question are -3 and 4
Now it is easy to imagine the three cases that x is to the left of -3, between -3 and 4 and to the right of 4. The only case when the two distances add up to the distance between -3 and 4, ie, 7 is case two. In case 1 and 3, the sum will exceed 7

1) could mean case 2 or 3. Not sufficient
2) could mean case 1 or 2. Not sufficient
1+2) can only mean case 2. Sufficient to know that y=7

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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27 Sep 2010, 00:47
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thirst4edu wrote:
If x & y are integers and y=|x+3| + |4-x|, does y equals 7?

1) x < 4
2) x > -3

Had a hard time solving this, would like to know how to solve this using number picking approach as well as algebraic approach. Thanks.

OA is
[Reveal] Spoiler:
C

Guys, the be low is my approcah for any modulus qtn in GMAT.

Remember.
The meaning of |x-y| is "On the number line, the distance of X from +Y"
The meaning of |x+y| is "On the number line, the distance of X from -Y"
The meaning of |x| is "On the number line, the distance of X from 0".

Qtn:

for integers X and Y, If y=|x+3| + |4-x|, does y equals 7
==> is the SUM of the distance b/e x and -3 , and x and 4 equals to 7?
==> .........-3......0...........4..... observe that X has to be anywhere b/w -3 and 4 or on any of these points for the total distance to be 7

Stmnt1: X < 4: Answer could be Yes if X is < 4 and b/w -3 and 4 but from the given information (i.e X<4) X could be some where left to -3 in which case the total distance would be > 7 hence insufficient.

......X....-3......0..........4 answer to the qtn: NO
or ............-3...X...0.........4 answer to the qtn: YES

Stmnt2: X > -3: Answer could be Yes if X is > -3 and b/w -3 and 4 but from the given information (i.e X>-3) X could be some where right to 4 in which case the total distance would be > 7 hence insufficient.
..........-3......0..........4...X... answer to the qtn: NO
or ..........-3...X...0.........4 answer to the qtn: YES

1&2

X must be b/w -3 and 4
..........-3.X.X.X...0.X.X.X.X.X...4...... answer is always YES..hence Sufficient.

Hope it helps

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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27 Sep 2010, 11:51
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thirst4edu wrote:
If x & y are integers and y=|x+3| + |4-x|, does y equals 7?

1) x < 4
2) x > -3

Had a hard time solving this, would like to know how to solve this using number picking approach as well as algebraic approach. Thanks.

OA is
[Reveal] Spoiler:
C

Either choice by itself is clearly insufficient:

(1) If x = 3, y = |3+3| + |4-3| = 7. If x = -100, then y = 97 + 104 = 201.
(2) If x = -2, y = 1 + 6 = 7. If x = 100, then y = 103 + 96 = 199.

Putting them together, you can quickly check every integer value of x from -3 to 4 and see that y = 7 for every one. It's only 6 values to check, you can do it very quickly in your head.

(C)

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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28 Sep 2010, 10:58
thanks bunuel. always great explanations!!

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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29 Sep 2010, 10:21
Bunuel rocks, cheers mate

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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26 May 2012, 08:38
At bunuel,

A. x<{-3} --> y=| x + 3| +|4-x| =-x-3+4-x=-2x+1, which means that when x is in the range {-infinity,-3} the value of y is defined by x (we would have multiple choices of y depending on x from the given range);

B. -3\leq{x}\leq{4} --> y=|x+3|+|4-x|=x+3+4-x=7, which means that when x is in the range {-3,4} the value of y is 7 (value of y does not depend on value of x, when x is from the given range);

C. x>{4} --> y=|x+3|+|4-x|=x+3-4+x=2x-1, which means that when x is in the range {4, +infinity} the value of y is defined by x (we would have multiple choices of y depending on x from the given range).

I understand the how you got the check points -3 and 4 but I am having a hard time understanding how to decide sign for x when you are removing absolute value symbol
for example for
(A) x<-3 y=| x + 3| +|4-x|
how did you decide sign of "x" here ===> -x-3+4-x
-2x+1

Similarly can u also explain for (B) and (C)

thank you!

-K

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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28 May 2012, 05:23
kartik222 wrote:
At bunuel,

A. x<{-3} --> y=| x + 3| +|4-x| =-x-3+4-x=-2x+1, which means that when x is in the range {-infinity,-3} the value of y is defined by x (we would have multiple choices of y depending on x from the given range);

B. -3\leq{x}\leq{4} --> y=|x+3|+|4-x|=x+3+4-x=7, which means that when x is in the range {-3,4} the value of y is 7 (value of y does not depend on value of x, when x is from the given range);

C. x>{4} --> y=|x+3|+|4-x|=x+3-4+x=2x-1, which means that when x is in the range {4, +infinity} the value of y is defined by x (we would have multiple choices of y depending on x from the given range).

I understand the how you got the check points -3 and 4 but I am having a hard time understanding how to decide sign for x when you are removing absolute value symbol
for example for
(A) x<-3 y=| x + 3| +|4-x|
how did you decide sign of "x" here ===> -x-3+4-x
-2x+1

Similarly can u also explain for (B) and (C)

thank you!

-K

Absolute value properties:
When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|\leq{-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|\leq{some \ expression}$$. For example: $$|5|=5$$;

So, for example if $$x<-3$$ then $$x+3<0$$ and $$4-x>0$$ which means that $$|x+3|=-(x+3)$$ and $$|4-x|=4-x$$ --> $$|x+3|+|4-x|=-(x+3)+4-x=-2x+1$$.

Similarly for B and C.

Hope it's clear.
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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28 May 2012, 09:10
Bunuel wrote:
thirst4edu wrote:
If x & y are integers and y=|x+3| + |4-x|, does y equals 7?

1) x < 4
2) x > -3

Had a hard time solving this, would like to know how to solve this using number picking approach as well as algebraic approach. Thanks.

OA is
[Reveal] Spoiler:
C

$$y=|x+3|+|4-x|$$ two check points: $$x=-3$$ and $$x=4$$ (check point: the value of $$x$$ when expression in || equals to zero), hence three ranges to consider:

A. $$x<{-3}$$ --> $$y=| x + 3| +|4-x| =-x-3+4-x=-2x+1$$, which means that when $$x$$ is in the range {-infinity,-3} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range);

B. $$-3\leq{x}\leq{4}$$ --> $$y=|x+3|+|4-x|=x+3+4-x=7$$, which means that when $$x$$ is in the range {-3,4} the value of $$y$$ is $$7$$ (value of y does not depend on value of $$x$$, when $$x$$ is from the given range);

C. $$x>{4}$$ --> $$y=|x+3|+|4-x|=x+3-4+x=2x-1$$, which means that when $$x$$ is in the range {4, +infinity} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range).

Hence we can definitely conclude that $$y=7$$ if $$x$$ is in the range {-3,4}

(1) $$x<4$$ --> not sufficient ($$x<4$$ but we don't know if it's $$\geq{-3}$$);
(2) $$x>-3$$ --> not sufficient ($$x>-3$$ but we don't know if it's $$\leq{4}$$);

(1)+(2) $$-3<x<4$$ exactly the range we needed, so $$y=7$$. Sufficient.

OR: looking at $$y=|x+3|+|4-x|$$ you can notice that $$y=7$$ ($$y$$ doesn't depend on the value of $$x$$) when $$x+3$$ and $$4-x$$ are both positive, in this case $$x-es$$ cancel out each other and we would have $$y=|x+3|+|4-x|=x+3+4-x=7$$. Both $$x+3$$ and $$4-x$$ are positive in the range $$-3<{x}<4$$ ($$x+3>0$$ --> $$x>-3$$ and $$4-x>0$$ --> $$x<4$$).

Hope it's clear.

inequalities are posing problems! - one doubt - when it is said "-3<x<4", in this range shouldn't we check -2, -1 or 2,1 etc and see what is the value for y?
is y independent of x when x<0?

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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01 Jun 2012, 05:36
Bunuel wrote:
thirst4edu wrote:
If x & y are integers and y=|x+3| + |4-x|, does y equals 7?

1) x < 4
2) x > -3

Had a hard time solving this, would like to know how to solve this using number picking approach as well as algebraic approach. Thanks.

OA is
[Reveal] Spoiler:
C

$$y=|x+3|+|4-x|$$ two check points: $$x=-3$$ and $$x=4$$ (check point: the value of $$x$$ when expression in || equals to zero), hence three ranges to consider:

A. $$x<{-3}$$ --> $$y=| x + 3| +|4-x| =-x-3+4-x=-2x+1$$, which means that when $$x$$ is in the range {-infinity,-3} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range);

B. $$-3\leq{x}\leq{4}$$ --> $$y=|x+3|+|4-x|=x+3+4-x=7$$, which means that when $$x$$ is in the range {-3,4} the value of $$y$$ is $$7$$ (value of y does not depend on value of $$x$$, when $$x$$ is from the given range);

C. $$x>{4}$$ --> $$y=|x+3|+|4-x|=x+3-4+x=2x-1$$, which means that when $$x$$ is in the range {4, +infinity} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range).

Hence we can definitely conclude that $$y=7$$ if $$x$$ is in the range {-3,4}

(1) $$x<4$$ --> not sufficient ($$x<4$$ but we don't know if it's $$\geq{-3}$$);
(2) $$x>-3$$ --> not sufficient ($$x>-3$$ but we don't know if it's $$\leq{4}$$);

(1)+(2) $$-3<x<4$$ exactly the range we needed, so $$y=7$$. Sufficient.

OR: looking at $$y=|x+3|+|4-x|$$ you can notice that $$y=7$$ ($$y$$ doesn't depend on the value of $$x$$) when $$x+3$$ and $$4-x$$ are both positive, in this case $$x-es$$ cancel out each other and we would have $$y=|x+3|+|4-x|=x+3+4-x=7$$. Both $$x+3$$ and $$4-x$$ are positive in the range $$-3<{x}<4$$ ($$x+3>0$$ --> $$x>-3$$ and $$4-x>0$$ --> $$x<4$$).

Hope it's clear.

MOD questions always floor me.
Could you please suggest some good material on MODs?

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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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01 Jun 2012, 07:37
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manulath wrote:
MOD questions always floor me.
Could you please suggest some good material on MODs?

Check Absolute Value chapter of Math Book: math-absolute-value-modulus-86462.html

DS questions on absolute value to practice: search.php?search_id=tag&tag_id=37
PS questions on absolute value to practice: search.php?search_id=tag&tag_id=58

Tough absolute value and inequity questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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20 Dec 2012, 09:01
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Thinking of modulus as distances

|x+3| => distance of x from -3
|x-4| => distance of x from 4

Picture the same on the number line

________-3_____________0_________________4__________

We are given that y is the sum of the distance of x from -3 & of x from 4

Hence y could be anywhere on the number line

For y=7, let us consider the possibilities

Case (1)

_____x______-3_____________0_________________4__________

As you can quickly conclude
Its impossible for the distance to be 7 if x < -3
Take x = -4 and check,
y = 1 + 8 = 9

Case (2)

_________-3_____________0_________________4_____x_____

As you can quickly conclude
Its impossible for the distance to be 7 if x >4
Take x = 5 and check,
y = 8 + 1 = 9

Hence the range for y = 7 has to be in third case

__-3_____________x_________________4_____
i.e. -3<x<4

So we need to find if -3<x<4 ????

(1) x < 4
Insuff

(2) x > -3
Insuff

(3) Combining - -3<x<4

Bang-on.

Hence C
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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27 Mar 2013, 07:58
Bunuel wrote:
thirst4edu wrote:
If x & y are integers and y=|x+3| + |4-x|, does y equals 7?

1) x < 4
2) x > -3

Had a hard time solving this, would like to know how to solve this using number picking approach as well as algebraic approach. Thanks.

OA is
[Reveal] Spoiler:
C

$$y=|x+3|+|4-x|$$ two check points: $$x=-3$$ and $$x=4$$ (check point: the value of $$x$$ when expression in || equals to zero), hence three ranges to consider:

A. $$x<{-3}$$ --> $$y=| x + 3| +|4-x| =-x-3+4-x=-2x+1$$, which means that when $$x$$ is in the range {-infinity,-3} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range);

B. $$-3\leq{x}\leq{4}$$ --> $$y=|x+3|+|4-x|=x+3+4-x=7$$, which means that when $$x$$ is in the range {-3,4} the value of $$y$$ is $$7$$ (value of y does not depend on value of $$x$$, when $$x$$ is from the given range);

C. $$x>{4}$$ --> $$y=|x+3|+|4-x|=x+3-4+x=2x-1$$, which means that when $$x$$ is in the range {4, +infinity} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range).

Hence we can definitely conclude that $$y=7$$ if $$x$$ is in the range {-3,4}

(1) $$x<4$$ --> not sufficient ($$x<4$$ but we don't know if it's $$\geq{-3}$$);
(2) $$x>-3$$ --> not sufficient ($$x>-3$$ but we don't know if it's $$\leq{4}$$);

(1)+(2) $$-3<x<4$$ exactly the range we needed, so $$y=7$$. Sufficient.

OR: looking at $$y=|x+3|+|4-x|$$ you can notice that $$y=7$$ ($$y$$ doesn't depend on the value of $$x$$) when $$x+3$$ and $$4-x$$ are both positive, in this case $$x-es$$ cancel out each other and we would have $$y=|x+3|+|4-x|=x+3+4-x=7$$. Both $$x+3$$ and $$4-x$$ are positive in the range $$-3<{x}<4$$ ($$x+3>0$$ --> $$x>-3$$ and $$4-x>0$$ --> $$x<4$$).

Hope it's clear.

Thanks for the great explanation . I got this answer when I tackled it using the same approach .. but got an E when I tried to tackle it in the +/- (x+3)=+/-(4-x) . I found y = +7 , -7 , 2x-1 , -2x+1 , and on plugging values that satisfy the 2 statements together it turned out to a range of -5 to 5 . Could you please guide ? Thanks a million
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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27 Mar 2013, 09:02
TheNona wrote:
Bunuel wrote:
thirst4edu wrote:
If x & y are integers and y=|x+3| + |4-x|, does y equals 7?

1) x < 4
2) x > -3

Had a hard time solving this, would like to know how to solve this using number picking approach as well as algebraic approach. Thanks.

OA is
[Reveal] Spoiler:
C

$$y=|x+3|+|4-x|$$ two check points: $$x=-3$$ and $$x=4$$ (check point: the value of $$x$$ when expression in || equals to zero), hence three ranges to consider:

A. $$x<{-3}$$ --> $$y=| x + 3| +|4-x| =-x-3+4-x=-2x+1$$, which means that when $$x$$ is in the range {-infinity,-3} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range);

B. $$-3\leq{x}\leq{4}$$ --> $$y=|x+3|+|4-x|=x+3+4-x=7$$, which means that when $$x$$ is in the range {-3,4} the value of $$y$$ is $$7$$ (value of y does not depend on value of $$x$$, when $$x$$ is from the given range);

C. $$x>{4}$$ --> $$y=|x+3|+|4-x|=x+3-4+x=2x-1$$, which means that when $$x$$ is in the range {4, +infinity} the value of $$y$$ is defined by $$x$$ (we would have multiple choices of $$y$$ depending on $$x$$ from the given range).

Hence we can definitely conclude that $$y=7$$ if $$x$$ is in the range {-3,4}

(1) $$x<4$$ --> not sufficient ($$x<4$$ but we don't know if it's $$\geq{-3}$$);
(2) $$x>-3$$ --> not sufficient ($$x>-3$$ but we don't know if it's $$\leq{4}$$);

(1)+(2) $$-3<x<4$$ exactly the range we needed, so $$y=7$$. Sufficient.

OR: looking at $$y=|x+3|+|4-x|$$ you can notice that $$y=7$$ ($$y$$ doesn't depend on the value of $$x$$) when $$x+3$$ and $$4-x$$ are both positive, in this case $$x-es$$ cancel out each other and we would have $$y=|x+3|+|4-x|=x+3+4-x=7$$. Both $$x+3$$ and $$4-x$$ are positive in the range $$-3<{x}<4$$ ($$x+3>0$$ --> $$x>-3$$ and $$4-x>0$$ --> $$x<4$$).

Hope it's clear.

Thanks for the great explanation . I got this answer when I tackled it using the same approach .. but got an E when I tried to tackle it in the +/- (x+3)=+/-(4-x) . I found y = +7 , -7 , 2x-1 , -2x+1 , and on plugging values that satisfy the 2 statements together it turned out to a range of -5 to 5 . Could you please guide ? Thanks a million

Could you please elaborate the red part?
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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27 Mar 2013, 09:37
Bunuel wrote:
Could you please elaborate the red part?

I mean that combining both statements x is between -3 and 4 , so I plugged all the values in this range in both 2x-1 and -2x+1 giving the range of Ys -5 to 5
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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28 Mar 2013, 11:42
TheNona wrote:
Bunuel wrote:
Could you please elaborate the red part?

I mean that combining both statements x is between -3 and 4 , so I plugged all the values in this range in both 2x-1 and -2x+1 giving the range of Ys -5 to 5

y is equal to 1-2x only when x<-3.
y is equal to 2x-1 only when x>4.

When -3<x<4, then y is equal to 7.
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7? [#permalink]

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07 May 2013, 21:08
mario1987 wrote:
Hi guys,
I would like to deeply understand how to deal with absolute value questions like the one attached.
Thank you very much

The Expression given in the Question is Y = lx+3l + l4-xl .. & Question asks Is Y = 7 ????

Statement 1 :: x<4 ..... when we plugin any value of x less than 4 till -3 we will get a result as Y = 7 but below -3 ... Y will not be equal to 7 . Therefore, Insufficient.

Similarly, Statement :: 2 .... is insufficient.......

with 1+2 ..... we get the value of x in between 4 & -3 ...... i.e., -3<x<4 ....... For all the value of x in between -3 & 4 ... the value of

Y is always. 7 .. Therefore, Sufficient....

Hence, C ...........
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Re: If x and y are integers and y = |x + 3| + |4 - x|, does y equals 7?   [#permalink] 07 May 2013, 21:08

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