emmak wrote:
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
VERITAS PREP OFFICIAL SOLUTION:Solution: C. Algebraically, the inequalities give you four different possibilities:
12x – 5 > 7 – 6x (which works out cleanly to 18x > 12, meaning x > 2/3)
12x – 5 < -(7 – 6x)
-(12x – 5) < 7 – 6x
-(12x -5) > -(7 – 6x)
With the last three all involving negatives and inequalities, it can be helpful to simply find the inequality point and then test values on either side to determine whether it's greater than or less than:
12x – 5 < -(7 – 6x) works out to:
12x - 5 < 6x - 7
6x < -2
x < −1/3, but try a value like -1 (less than −1/3) and like 0 (greater than −1/3) to ensure the - vs. + of a tricky inequality with absolute values. -1 fits with the given information and 0 does not, so it should be clear that x < −1/3.
-(12x – 5) < 7 – 6x works out to 5 - 12x < 7 - 6x, which gives you -2 < 6x, and x > −1/3. This is why testing negative/positive is so important...the four "original" inequalities allow for really all sets of possible values other than −1/3 and 2/3, so some quick trial and error can help you determine which side of the inequalities are valid and which are not. Again, it should be clear from a quick plug-in of 0 and -1 that x < −1/3.
-(12x -5) > -(7 – 6x) works out to 5 - 12x > 6x - 7, which leads you to 12 > 18x, and x < 2/3. This confirms the "break point" of 2/3, so again a quick plug in of easy numbers on either side (0 and 1) will help you determine that x must be greater than 2/3.
So you know that x is either greater than 2/3 or less than −1/3. Certain answer choices, then, are easy to pick off: -12 could be -1*12. 17 could be 1*17. 4/9 could be -1*(-4/9) and -7/5 could be -1*7/5. But -2/9 cannot be done.