Problem Solving

If |12x−5|>|7−6x|, which of the following CANNOT be the product of two possible values of x
?
A) −12
B) -7/5
C) -2/9
D) 4/9
E) 17

OA : C
Source : Veritas Prep

I can work out the four different inequalities that can be derived from the question to get 4 different ranges of X, but how to proceed from there?

If |12x-5|>|7-6x|, which of the following CANNOT be the product of two possible values of x?

a -12
b -7/5
c -2/9
d 4/9
e 17

Determine the CRITICAL POINTS: the values where |12x-5| = |7-6x|.
There are TWO CASES to consider:
Case 1: The signs of the equation are UNCHANGED.
Case 2: For ONE SIDE of the equation, the signs are flipped.

Case 1: 12x-5 = 7-6x
18x = 12
x = 12/18 = 2/3.

Case 2: 12x-5 = -7+6x
6x = -2
x = -2/6 = -1/3.

The CRITICAL POINTS are -1/3 and 2/3.
These are the only values where |12x-5| = |7-6x|.
To determine the ranges where |12x-5| > |7-6x|, test one value to the left and right of each critical point.

x<-1/3:
Plug x = -1 into |12x-5| > |7-6x|:
|12(-1) - 5| > |7 - 6(-1)|
17 > 13.
This works.
x < -1/3 is a valid range.

-1/3 < x < 2/3:
Plug x = 0 into |12x-5| > |7-6x|:
|12*0 - 5| > |7 - 6*0|
5 > 7.
Doesn't work.
-1/3 < x < 2/3 is NOT a valid range.

x > 2/3:
Plug x = 1 into |12x-5| > |7-6x|:
|12*1 - 5| > |7 - 6*1|
7 > 1.
This works.
x > 2/3 is a valid range.

Thus, |12x-5| > |7-6x| when x < -1/3 or x > 2/3.
Eliminate answer choices that indicate a possible product.
If the two solutions are x=-1 and x=12, then their product = -1*12 = -12.
Eliminate A.
If the two solutions are x=-1 and x=7/5, then their product = -1 * 7/5 = -7/5.
Eliminate B.
If the two solutions are x=-1 and x=-4/9, then their product = -1 * -4/9 = 4/9.
Eliminate D.
If the two solutions are x=1 and x=17, then their product = 1*17 = 17.
Eliminate E.

The correct answer is C.