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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.

Answer: C.

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).

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

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.

(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.

Re: If x and y are integers and y=|x+3| + |4-x|, does y equals 7 [#permalink]
26 May 2012, 07:38

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

Re: If x and y are integers and y=|x+3| + |4-x|, does y equals 7 [#permalink]
28 May 2012, 04:23

Expert's post

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.

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.

Answer: C.

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?

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.

Answer: C.

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?

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.

Answer: C.

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
_________________

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.

Answer: C.

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?
_________________

Re: Power Prep - DS - Modulus [#permalink]
27 Mar 2013, 08:37

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
_________________

Re: Power Prep - DS - Modulus [#permalink]
28 Mar 2013, 10:42

Expert's post

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.
_________________

Re: Algebra->Inequalities (Absolute Value) [#permalink]
07 May 2013, 20:08

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 ...........
_________________

If you don’t make mistakes, you’re not working hard. And Now that’s a Huge mistake.

gmatclubot

Re: Algebra->Inequalities (Absolute Value)
[#permalink]
07 May 2013, 20:08