True or False: | 3x - 9 | < 7 and -2/3 < x < 16/3 are both equivalent statements?

Question

True ?

Answer 1

You both are wrong. Namely,

3x-9, 3x-9>=0 <=> x >=3
|3x-9|=
-3x+9, x < 3

So, when x < 3,
|3x-9| < 7 <=> -3x + 9 < 7 <=> -3x < -2 <=> x > 2/3
So, we have: (x < 3) and (x > 2/3). therefore, the solution for first case is: 2/3 < x < 3

when x >= 3
|3x-9|<7 <=> 3x - 9 < 7 <=> 3x < 16 <=> x < 16/3. Solution:
3 <= x < 16 / 3

Konjunction of those two cases is:

x e (2/3,3) U [3,16/3), or 2/3 < x < 16/3

*when i said both, i meant poster and first response

Answer 2

Suppose x>3 Then 3x - 9 < 7. Then x < 16/3
Suppose x = 3 then 0 < 7 is true
Suppose x <3 then -(3x - 9) < 7. Then x > 2/3
The solution of | 3x - 9 | < 7 is 2/3 < x < 16/3
and that is not equivalent to -2/3 < x < 16/3.
So the statement is false.

Th

Answer 3

NOT equivalent. BUT, both TRUE!