Problem Solving

If x < 12, then it must be true that

(a) -x < -12
(b) -x - 2 < 14
(c) -x + 2 < -10
(d) x + 2 < 10
(e) x - 2 < 11

OA is E but can't understand why.

Please explain.

Let's take each inequality at a time:
A. -x < -12 - multiply by (-1) on each side to eliminate the minuses and you get:
x > 12 - "<" gets reversed when you multiply by a negative number. So the initial inequality is equivalent to x > 12, which is not consistent with x < 12.

B. -x - 2 < 14 - add 2 on each side to get;
-x < 16 - multiply by (-1)
x > - 16. While it is true that for some numbers that are smaller than 12 this is correct, it isn't for everything. Just consider x = 20:
20 > -16, but 20 is not greater than 12.

C. -x + 2 < -10 - subtract 2 from each side
-x < - 12 - takes us back to A

D. x + 2 < 10 means that x < 8. Again, take x = 10. While it is true that x < 12, 10 is not smaller than 8.

E is our answer: x - 2 < 11 means that x < 13. Now, this is consistent with x < 12, since any number smaller than 12 will also be smaller than 13.

But if x< 13, x can be equal to 12. And if x = 12, x cannot be less than 12.

For answer D, however, where it can be derived that x < 8, x can be any number less than 8 (i.e. 7, 6, 5, etc). And if this is the case, then this statement MUST be true?

Read the question slowly and then DanaJ's answer. You'll get it.

D. x + 2 < 10 means that x < 8. Again, take x = 10.
While it is true that x < 12, 10 is not smaller than 8.

A number smaller than 12 will be smaller than 13.