For QUESTION ONE: Is that a < or a > between the absolute values? Regardless, you'll need to split it into FOUR inequalities:
Just like
| 3x - 2 | > 10
would become
3x - 2 > 10
AND
3x - 2 < -10
If you double the absolute values, you're going to have four inequalities instead of two.
For QUESTION TWO:
s - 1 / s < 1 / t - t
Is this correct? t - t = 0, which will be undefined since it's in the denominator. Just want to make sure there's no error here in the way you've copied it....
For QUESTION THREE:
x /12 > y / 40? Yes/no DS.
To determine sufficiency, we need to know the value of x AND the value of y.
(1) Gives us the range for the relationship between x and y.
Let's say x = 3, then the inequality would become 30 < 3y - 6, or 12 < y. Since y can be anything as long as it is greater than 12, we could say y = 20. That would make our inequality 3/12 > 20/40, or 1/4 > 1/2 which is FALSE.
Let's say x = -3, then the inequality would become -30 < 3y - 6, or -24 < 3y, or -8 < y. Again, no matter what we choose, it will make our original inequality FALSE.
Let's say x = 0, then the inequality would becomes 0 < 3y - 6, or 6 < 3y, or 2 < y. Let's say y = 40. Then we'd get 0/12 > 40/40 for our original inequality. That becomes 0 > 1, another FALSE statement.
We can see that (1) is sufficient, since when we choose very different sets of numbers, we answer our original question in the same way.
(2) Gives another range for the relationship between x and y. Let's try some numbers again to test the results.
If x = 0, the inequality becomes 12(0) - 7 > 4y, then 0 - 7 > 4y, then -7/4 > y. Let's say y = -2. Plugging those back in to the original inequality: 0 > -1/20. This is a TRUE statement.
If x = -1, the inequality becomes 12(-1) - 7 > 4y, then -12 - 7 > 4y, then -19 > 4y, then -19/4 > y. Let's say y = -5. The original inequality becomes -1/12 > -5/40, or -1/12 > -1/8. This is another TRUE statement.
If x = 1/2, then 12(1/2) - 7 > 4y, then 6 - 7 > 4y, then -1/4 > y. Let's say y = -1/2. Plugging that into the original inequality: (1/2)/12 > (-1/2)/40, becomes 1/2 x 1/12 > -1/2 x 1/40, becomes 1/24 > -1/80. Again, a TRUE statement.
IMO, the answer is [spoiler](C)[/spoiler].
