12 Days of Christmas GMAT Competition - Day 8: What is the value of

What is the value of |2x−5|+|4x+3|−|x–9|?

(1) (x−5)/(x−3)<0
(2) 4x−|x|=16–x

(1)
3<x<5

Not sufficient. We cannot assume x is an integer

(2)

4x - x = 16 - x

x = 4

x cannot be negative because RHS will be positive and LHS negative

|2x−5|+|4x+3|−|x–9|
= |3|+|19|-|5| = 17

Sufficient

B is the answer

What is the value of |2x−5|+|4x+3|−|x–9|?

(1) (x−5)/(x−3)<0
Either x-5< 0 and x-3>0 so x<5 and x> 3
or x-5> 0 and x-3<0 so 3<x<5
So we open the expression for this range: 2x-5+4x+3+x-9=7x-11. We dont know x thus not sufficient


(2) 4x−|x|=16–x
5x-x=16 when x>0
x=4

or

5x+x=16 when x<0
x=16/6 not possible because x<0
Thus x=4 Sufficient

Ans B

(1) On solving 1, 3<x<5

also,

the question leads to 7x-11

Answer can't be found as 3<x<5

(2) On solving 1, x=4

the question.. in this prompt also... leads to 7x-11

Answer can be found as x=4; hence, 7x-11=17

IMO, (B) will be the answer

In order to find this value, we need to know the only possible value for X - because even if we know the range of X, we will be able to open the modulus,
but will still not know to what the whole expression equals.

So, let's look at the options:
This gives us the fact that one of these brackets must be below zero, while the other is above. The only case when this happens is when X is above 3 but below 5:
\(3 < X < 5\)
And while this does give us the necessary range to open the modulus in the original expression, we still don't know the precise value, as X can be 3.1 as well as 4.9, for instance.
Therefore, Condition 1 by itself is insufficient - it would be only if we knew that X is integer, which we don't.

We have two scenarios here:

• \(x < 0 \) and \(4x - (-x) = 16 - x\)
  So, \(5x = 16 - x\)
  \(x = \frac{16}{6}\) and \(x<0\) - which is impossible, so this is not a solution
• \(x >= 0\) and \(4x - x = 16 - x\)
  So, \(4x = 16, x = 4\) and \(x>0\) - which works just fine.

Therefore, there is only one possible solution, and it is enough to resolve the main expression.
Thus, Condition 2 by itself is sufficient.

The answer is B.

What is the value of |2x−5|+|4x+3|−|x–9| ?

(1) (x−5)/(x−3) < 0
The range of X in this condition is 3 < x < 5

In this range X can have many possible values. Hence, Statement 1 is Insufficient.

(2) 4x−|x|=16–x
In this case the value of X is 4 since −|x| will always be –x.

Since we have unique vale of X, therefore we can find the value of |2x−5|+|4x+3|−|x–9| .
Hence, Statement 2 is Sufficient.

Answer: Option B

What is the value of |2x−5|+|4x+3|−|x–9|?

We need to find the value of x.
(1) \(\frac{(x−5)}{(x−3)}\)<0
either the numerator or denominator is negetive
On solving for x, we have
3<x<5

A definite value of x cannot be defined as x can be an integer or a fraction
Insufficient

(2) 4x−|x|=16–x
solving for x, we have
4x-x=16-x
4x=16
x=4

4x+x=16-x
6x=16
x=8/3

checking if 4x−|x|=16–x holds good for the values of x=4 and x=8/3
only x=4 holds
Hence for x=4 the value of |2x−5|+|4x+3|−|x–9| can be obtained.
Sufficient

Ans : B

(1), leads to

3<x<5, which furthjer leads to

non-constant value since different values can be found depending on the value of x

So, Insufficient

(2), leads to

x=5, which further leads to

constant value since only one value can be found for x=4

So, Sufficient

Thus (B) will be the correct option

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