3. If 2 < x < 6, which of the following statements about x are necessarily true, and which are not necessarily true?

a) 0 < x < 4

b) 0 < x-2 < 4

c) 1 < x/2 < 3

d) 1/6 < 1/x < 1/2

e) 1 < 6/x < 3

I know what they're asking. They're asking which ones are true and which ones aren't. But the 2 < x < 6?

The answer is not necessarily true (saw it in the back of the textbook but there aren't any explanations as to why this is so).

Is it because:

$\displaystyle 2 < x < 6$

$\displaystyle 0 < x < 4$

Well it would be x > 2, so x > 0 would be true. And then the next part x < 6 but x < 4 could not be true because the answer could be 4 or 5?

And for this one:

$\displaystyle 2 < x < 6$

$\displaystyle 0 < x -2 < 4$

x > 2

x > 0

Well that wold be true because if x > 2, then x is definitely greater than 0. And if x <6 and x-2 < 4 would mean that x would have to be at least 5 since x is less than 6 and if you use 5-2 = 3 <4, that holds true.

And then this one:

$\displaystyle 2 < x < 6$

$\displaystyle 1 < x/2 < 3$

x is greater than 2, so 1/2x is greater than 1. Well I think that's true because if you have x greater than 2 then you can try 3 immediately and half of 3 is 1.5, which is greater than one. And if x <6, then the highest you can half is 5 and 1/2(5) = 2.5, which is less than three. This would be true.

$\displaystyle 2 < x < 6$

$\displaystyle 1/6 < 1/x < 1/2$

I don't know how to approach this problem now. I'll try though...

If x > 2 then on the bottom we can put in 1/3... and 1/3 is greater than 1/6th. 1/3 would be less than 1/2. Plug in 4... 1/4 is greater than 1/6 and 1/4 is less than 1/2. And since we're restricted to x < 6, the last value we plug in will be 5 to see if this is true...

1/6 < 1/5< 1/2.

1/5>1/6 and 1/5 < 1/2. So this true.

$\displaystyle 2 < x < 6$

$\displaystyle 1 < 6/x < 3$

x > 2 so 6/x > 1. Plug in 3 = 6/3=2 > 1. 6/5 = 1.2 > 1...so this is true too.

Am I doing this right?

There's one here that I didn't write above but it's an absolute value problem.

So...

$\displaystyle 2 < x < 6$

$\displaystyle |x-4| < 2$

x > 2 which means 3-4 = -1 and the |-1| = -(-1) = 1 < 2.

Take it up x being 5 and |5-4| = 1 absolute value is, so 1 < 2. This would be true as well.

$\displaystyle 2 < x < 6$

$\displaystyle -6 < -x < 2$

Since x has to be greater than 2, we can plug a 3 in. That would make it -3 > -6. True. Now -3 < 2 is true. And from the original problem it has to be x > 2 and x>6. So we're only using the values of 3, 4, and 5.

Plug in 5.

-5>-6. True.

-5< 2. True.

This is true.

Am I doing this all right?