Problem Solving

dzelkas wrote:Is t a non integer?

(1) |t-3|<1
(2) t(does not not = )3

I understand that it's not (2)

but I am confused on (1) only because this is the solution:

|t-3|<1
-1<t<-3 can someone explain this concept to me
2<t<4 and then this part, where i got lost as well.


shouldnt it be
|t-3|<1
t<4???

any help appreciated

thanks

danny

Whitney Garner wrote:

dzelkas wrote:but I am confused on (1) only because this is the solution:

|t-3|<1
-1<t<-3 can someone explain this concept to me
2<t<4 and then this part, where i got lost as well.

shouldnt it be
|t-3|<1
t<4???

Hi Danny!

Let's first attack the problem of solving an inequality when there is an absolute value. This should help the confusion you mentioned:

If we need to solve the expression |t-3| < 1, we first need to "jailbreak" that (t-3) from the absolute value symbols. To do so, we need to consider that 1 of 2 things may be true.

(A) (t-3) might be positive; therefore the absolute value symbols aren't really doing anything.

(B) (t-3) is actually negative; therefore the absolute value symbols are working to change the sign of (t-3).

Okay, so let's take (A) since it's easier. If the absolute value symbols aren't doing anything, we can just remove them and solve:

(t-3) < 1
t < 4

Now, we need to address scenario (B). If t-3 is actually negative, then if we throw away the absolute value symbol, WE need to manually change the sign on (t-3). We can accomplish that by just multiplying it times a -1.

-1*(t-3) < 1
-t + 3 < 1
2 < t

So, if we put A and B together, we get the proper range of values for |t-3|<1 = 2 < t < 4 (test some values to check...if I plug in 2 and 4, I see that it equals 1, and if I pick stuff in between (2.5, 3, 3.5, etc) I see that it works!

Now, let's get back to the original problem:

Original Question wrote:Is t a non integer?
(1) |t - 3| < 1.
(2) t ≠ 3.

Statement (1) only tells us that 2<t<4, but doesn't guarantee we get an integer! NOT Sufficient

Statement (2) only tells us that t≠3; it could therefore be any other possible integer or non-integer in the world! NOT Sufficient

Statement (1+2) = now we know that t must be a value between (but not including) 2 and 4. Well, the only integer in that range is 3, but statement 2 says it cannot be 2...Therefore we know that it MUST be a NON-integer! Sufficient

Hope this clears that up a bit!

Whit