Problem Solving

If |a - b| = 6 and |b - c| = 15, then what is the value of |c|?

(1) |a - c| = 9
(2) |b| = 9

Source: Empowergmat

Solution

Fact 2) |b| = 9 ..........it means that b=9 or b=-9

When b= 9

|9 - c| = 15 ............ C= -6 or C= 24

|-9 - c| = 15 ............ C= 6 or C= -24

Insufficient


Fact 1) |a - c| = 9

As both facts do not contradict each other, we can use the info of fact 1



When b=9 & From |a - b| = 6...............a = -3 or 15

When c=24 & Using fact 1.... |a - c| = 9 .........a = 33 or 15

Then a =15

When b=-9, c=6

When b=-9 & From |a - b| = 6...............a = -3 or -15

When c=6 & Using fact 1.... |a - c| = 9 .........a = -3 or 15

Then a =-3

We have two sets that stratifies 3 equations

Insufficient


Combined 1 & 2, No certain answer reached

Answer: E

Is my solution valid? Is there any easier solution

Thanks

Mo2men wrote:If |a - b| = 6 and |b - c| = 15, then what is the value of |c|?

(1) |a - c| = 9
(2) |b| = 9

|x-y| = the DISTANCE between x and y.
According to the prompt:
The distance between a and b is 6.
The distance between b and c is 15.
According to Statement 1:
The distance between a and c is 9.

Statement 2:
Let b=9.
Since this problem involves multiple DISTANCES, plot options on a NUMBER LINE:


Case 1:
c=-6 and a=3 satisfy the constraints that |a - b| = 6 and |b - c| = 15.
These values also satisfy Statement 1, since |a - c| = 9.
In this case, |c| = 6.

Case 2:
a=15 and and c=24 satisfy the constraints that |a - b| = 6 and |b - c| = 15.
These values also satisfy Statement 1, since |a - c| = 9.
In this case, |c| = 24.

Since Cases 1 and 2 satisfy both statements but yield different values for |c|, the two statements combined are INSUFFICIENT.

The correct answer is E.