If x is an integer, is |2x - 5| + |3x - 10| = 5 - x ?

Question

TOUGH QUESTION

(1) x is divisible by 2
(2) x is positive

chetan2u · Answer

Either you solve or apply logic...

|2x - 5| + |3x - 10| = 5 - x ..
Now,  5-x\geq{0}....x\leq{5} 
 If x is a negative integer, the right hand side will increase by ONE time, while left hand side will increase by FIVE (2x+3x) times as all terms within mod on LHS get added. 
 So x can be 0, 1, 2, 3, 4, 5..

A quick check by substituting 0, 1, 2, 3,4 ,5 in equation |2x - 5| + |3x - 10| = 5 - x tells us that x can be only 3..
 |2*3 - 5| + |3*3 - 10| = 5 - 3......|6-5|+|9-10|=21+1=2 .. 
  So, the question is finally IS x=3

(1) x is divisible by 2
So ans is No...

(2) x is positive
x can be 2, 3 or so on

A..

Other way would be to solve for equation

lacktutor · Answer

|2x—5| + |3x—10|= 5–x

1) x<=5/2
-2x+ 5–3x+10=5–x
—4x =—10
x= 5/2 ( satisfies the equation)

2) 5/2< x <10/3
2x—5 —3x +10=5–x
0=0

3) x>=10/3
2x—5 +3x—10=5–x
6x=20
x= 10/3 (satisfies the equation)

The question is that Are x=5/2 and x=10/3 ???

Statement1: x is divisible by 2.
—> both values of x are not divisible by 2.
—> The answer is Always No.
Sufficient

Statement2: x is positive.
Yes. Both answers are positive numbers.
Sufficient

The answer is D.

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itoyj · Answer

Question stem says "x is an Integer"

satya2029 · Answer

Here is my approach

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alexisrakh · Answer

|2x - 5| + |3x - 10| = 5 - x

Note that this equation implies that 2x - 5>=0 and 3x - 10<=0

Thus, 5/2<=x and x<=10/3, the only integer value that satisfies both inequalities is 3.

(1) the answer is always no. (Sufficient)
(2) x positive implies we could go either way, yes if x=3, and no if x is different from 3 (Not sufficient)

Ans. A

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shabuzen102 · Answer

Dear  chetan2u ,

Would you please explain why lacktutor's answer is wrong? He seems to be doing exactly what you said in your Absolute modulus - critical points and dividing number line into different regions. How come x can be 3 but not 5/2 or 10/3 (let's ignore the condition of x being an integer for now). If this were a normal absolute value question, something like what is value of x (and x doesn't have to be an integer), then what would be the answer?

Thank you!

chetan2u · Answer

Look at the condition 2...5/2<x<10/3
It gives 5-x=5-x or 0=0

Generally you wouldn’t get such a situation but here it would mean that all values in this range would fit in. 
In this range there is ONLY one integer and that is 3. 
So again the question boils down to is x=3

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If x is an integer, is |2x - 5| + |3x - 10| = 5 - x ?

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