If x is an integer, then what is the minimum possible value of |x| + |

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Competition Mode Question

If x is an integer, then what is the minimum possible value of  |x| + |x + 3| + |x + 6| + |x + 9| + |x + 12| + |x + 15| + |x + 18| ?

A. 63
B. 36
C. 32
D. 30
E. 25

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chetan2u · Accepted Answer

|x| + |x + 3| + |x + 6| + |x + 9| + |x + 12| + |x + 15| + |x + 18| ?

x should be negative and equal to the average of all the numeric values..
so  x=-\frac{0+3+6+9+12+15+18}{7}=-\frac{63}{7}=-9 
So, |-9| + |-9 + 3| + |-9 + 6| + |-9 + 9| + |-9 + 12| + |-9 + 15| + |-9 + 18|=9+6+3+0+3+6+9=36

debjit1990 · Answer

Ans: B

(x)+x+3+x+6+x+9+x+12+x+15+x+18=0
7x+63=0,x=-9
putting -9 in the given equation

value is: 36

NitishJain · Answer

B.

I tried solving it using deviation.
Since it is mod, the result will always be +ve. when plotting on number line for 0, -3. -6, -9, -12, -15, - 18, if we put x =-9, it will have minimum deviation and hence minimum value.

ArunSharma12 · Answer

If x is an integer, then what is the minimum possible value of |x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18|?

A. 63
B. 36
C. 32
D. 30
E. 25

to get to a minimum value, consider the median term and reduce it to zero. This will result in numbers with lowest dispersion and overall sum will be minimum
x =-9
sum = 9 + 6 + 3 + 0 + 3 + 6 + 9 = 18 + 12 + 6 = 36
Ans: B

NiftyNiffler · Answer

If x is an integer, then what is the minimum possible value of |x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18||x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18|?

A. 63
B. 36
C. 32
D. 30
E. 25

Min value of x = -9. To minimize the value, we need the middle modulus 0 so, the differences get equally distributed.

So, 9 + 6 + 3 + 0 + 3 + 6 + 9 = 36 (B)

freedom128 · Answer

the minimum possible value of |x| + |x+3| + |x+6| + |x+9| + |x+12| + |x+15| + |x+18| logically occurs when -18 <= x <= 0.

Try x=-9... 9 + 6 + 3 + 0 + 3 + 6 + 9 = 36

Try x=-6... 6 + 3 + 0 + 3 + 6 + 9 + 12 = 39. We get same result when x=-12

FINAL ANSWER IS (B) 36

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unraveled · Answer

If x is an integer, then what is the minimum possible value of |x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18| ?

A. 63
B. 36
C. 32
D. 30
E. 25
Here |x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18| would always be ≥ 0. 
To get the minimum possible sum of these seven terms x < 0

As |x|+|y|≥|x+y|
So, |x+y| can take value at least equal to |x|+|y|
|x+y| = |x|+|y|

Hence
|x| + |x+3| + |x+6| + |x+9| + |x+12| + |x+15| + |x+18| 
|x| + |x| + |3| + |x| + |6| + |x| + |9| + |x| + |12| + |x| + |15| + |x| + |18| = 7|x| + 63(suppose 'y')
Therefore minimum possible value would be less than 63.

Thus, -7x + 63 ≤ y ≤ 7x + 63 
Since minimum possible value of -7x + 63 = 0
x = -9

Therefore, |x| + |x+3| + |x+6| + |x+9| + |x+12| + |x+15| + |x+18| = |-9| + |-9+3| + |-9+6| + |-9+9| + |-9+12| + |-9+15| + |-9+18|
= 9 + 6 + 3 + 0 + 3 + 6 + 9 = 36

Answer B.

exc4libur · Answer

for x=-3, all decrease except for |x|
for x=-6, all decrease except |x| |x+3| 
for x=-9, all decrease except |x| |x+3|
for x=-12, all decrease except |x| |x+3| |x+6|

test -9 since its impact is greater than -6
x=-9: 9+6+3+0+3+6+9=18+18=36

ans (B)

rendezvous · Answer

Minimum @ position 4, x=-9, replacing x with 9 & solve: 36

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HoneyLemon · Answer

B could be the answer . 
 Below is my reasoning .

x is an integer, 
Min of 
|x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18||x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18|?

We need to find the minimum possible value of sum of the distance of X from 0 , -3 , .... -18. 
 If X is exactly midway of 0 and -18 i.e if x= -9 the sum distance will of x from respective points will be minimal and symmetric. 
So at X=-9 
 The sum of the distances are 
 |x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18||x|+|x+3|+|x+6|+|x+9|+|x+12|+|x+15|+|x+18|?
= 9 + 6 + 3 + 0+ 3 + 6 + 9 = 36 .

bidskamikaze · Answer

The mean of the 7 terms in AP (0,3,6,9,12,15,18) is the 4th term = 9.
The minimum value will occur at x=-9.
So we get 9+6+3+0+3+6+9 = 36
Ans B (36)

pooja.sharma2909 · Answer

Can someone tell me what will be the maximum value for the question?

Bunuel · Answer

The expression's maximum value is unbounded. As x approaches positive or negative infinity, the value of the expression correspondingly approaches infinity.

Surbhi23 · Answer

All values are inside modulus and hence, all of them are positive. 
The minimum value of modulus can be 0. But the sum = 0 is not possible here. Also, you can have just one term equal to zero here. If X=0, first term becomes 0 but other terms are still there. 
If X=-3 then except for second term, all other terms give some good positive value. 
Let’s make the largest contributing term =0 I.e. X=-18 but because of presence of so many modulus in equations, first, second and third terms are quite big. 
Notice that the terms differ by 3. It’s kinda an AP like situation. Also, there are odd number of terms I.e. 7. So why not make the middle term equal to zero and then on both sides, simplification can give us least values out of modulus? 
Let’s make 4th term =0 I.e. X=-9 
Expression becomes
9+6+3+0+3+6+9 = 36

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If x is an integer, then what is the minimum possible value of |x| + |

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If x is an integer, then what is the minimum possible value of |x| + |

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