Find all possible values for the inequality.

Question: What are the possible values of |2x−3| when 0<|x−1|<2?

My solution:

We know ∣∣x−1∣∣

becomes x-1 if x-1≥0 and -(x-1) if x-1<0.

Now consider two cases.

Case 1:

0<x-1<2 ⇒

1<x<3 ⇒

-1<2x-3<3.

Case 2:

0<-x+1<2 ⇒

-1<x<1 ⇒

-2<2x<2 ⇒

-5<2x-3<-1

Then the possible value include |2x-3|<3. Am I solving this problem correctly ??

Note, -5<|2x-3|<-1 would not work since the |2x-3| is bounded between two negative values.

Am I solving this problem correctly ?? The solution seems so incomplete?

Re: Find all possible values for the inequality.

If 0< |x- 2|< 2 then -2< x- 2< 2 (with ). Adding 2 to each part, 0< x< 4 (with ). Multiplying by 2, 0< 2x< 8 (with ). Subtracting 3, -3< 2x- 3< 5 (with 2x- 3\ne 1). That is, if 0< |x- 2|< 2, 2x- 3 can be any number strictly between -3 and 5, except 1.

Re: Find all possible values for the inequality.

Re: Find all possible values for the inequality.

Hello, turbozz!

Re: Find all possible values for the inequality.

Quote:

Originally Posted by

**HallsofIvy** If 0< |x- 2|< 2 then -2< x- 2< 2 (with

). Adding 2 to each part, 0< x< 4 (with

). Multiplying by 2, 0< 2x< 8 (with

). Subtracting 3, -3< 2x- 3< 5 (with 2x- 3\ne 1). That is, if 0< |x- 2|< 2, 2x- 3 can be any number strictly between -3 and 5, except 1.

I don't understand why your using 0<|x-2|<2 ?? How does this relate to the inequality at hand?

Re: Find all possible values for the inequality.

Quote:

Originally Posted by

**Soroban** Hello, turbozz!

how to do go from 0 <|x-1|<2 to -2<x-1<2 ? And x=1 wouldn't work since it wouldn't satisfy 0<|x-1|<2.

Re: Find all possible values for the inequality.

Quote:

Originally Posted by

**turbozz** Question: What are the possible values of |2x−3| when 0<|x−1|<2? My solution:

We know ∣∣x−1∣∣

becomes x-1 if x-1≥0 and -(x-1) if x-1<0. Now consider two cases. Case 1: 0<x-1<2 ⇒

1<x<3 ⇒

-1<2x-3<3. Case 2: 0<-x+1<2 ⇒

-1<x<1 ⇒

-2<2x<2 ⇒

-5<2x-3<-1

Then the possible value include |2x-3|<3. Am I solving this problem correctly ??

Note, -5<|2x-3|<-1 would not work since the |2x-3| is bounded between two negative values.

Am I solving this problem correctly ?? The solution seems so incomplete?

The triangle inequality works nicely here to get an upper bound:

So

So we can say .

Also, the reverse triangle inequality is useful here to get a lower bound. . So that means

So that means we have .

Re: Find all possible values for the inequality.

Re: Find all possible values for the inequality.

Quote:

Originally Posted by

**Prove It** The triangle inequality works nicely here to get an upper bound:

So

So we can say

.

Also, the reverse triangle inequality is useful here to get a lower bound.

. So that means

So that means we have

.

I like the way you showed the solution a lot. Only one issue though shouldn't 0<|2x-3|<5 ?

Re: Find all possible values for the inequality.

Quote:

Originally Posted by

**turbozz** Only one issue though shouldn't 0<|2x-3|<5 ?

Did you read reply #8?

The answer is .

Look at this: and .