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 $\displaystyle x- 2\ne 0$). Adding 2 to each part, 0< x< 4 (with $\displaystyle x\ne 2$). Multiplying by 2, 0< 2x< 8 (with $\displaystyle 2x\ne 4$). 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!

Quote:

$\displaystyle \text{What are the possible values of }|2x-3|\,\text{ when }\,0 < |x-1| < 2\,?$

$\displaystyle \begin{array}{ccccccc}\text{Given: } & 0 & < & |x-1| & < & 2 \\ \\ \text{Then:} & \text{-}2 & < & x-1 & < & 2 \\ \\ \text{Add 1:} & \text{-}1 & < & x & < & 3 \\ \\ \text{Times 2} & \text{-}2 & < & 2x & < & 6 \\ \\ \text{Minus 3:} & \text{-}5 &<& 2x-3 &<& 3 \end{array}$

Re: Find all possible values for the inequality.

Quote:

Originally Posted by

**HallsofIvy** If 0< |x- 2|< 2 then -2< x- 2< 2 (with $\displaystyle x- 2\ne 0$). Adding 2 to each part, 0< x< 4 (with $\displaystyle x\ne 2$). Multiplying by 2, 0< 2x< 8 (with $\displaystyle 2x\ne 4$). 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!

$\displaystyle \begin{array}{ccccccc}\text{Given: } & 0 & < & |x-1| & < & 2 \\ \\ \text{Then:} & \text{-}2 & < & x-1 & < & 2 \\ \\ \text{Add 1:} & \text{-}1 & < & x & < & 3 \\ \\ \text{Times 2} & \text{-}2 & < & 2x & < & 6 \\ \\ \text{Minus 3:} & \text{-}5 &<& 2x-3 &<& 3 \end{array}$

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: $\displaystyle \displaystyle \begin{align*} |a + b| \leq |a| + |b| \end{align*}$

So $\displaystyle \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\leq |2x - 2| + |-1| \\ &= 2|x - 1| + 1 \\ &< 2(2) + 1 \textrm{ since } |x - 1| < 2 \\ &= 5 \end{align*}$

So we can say $\displaystyle \displaystyle \begin{align*} |2x - 3| < 5 \end{align*}$.

Also, the reverse triangle inequality is useful here to get a lower bound. $\displaystyle \displaystyle \begin{align*} \left| |a| - |b| \right| \leq |a - b| \end{align*}$. So that means

$\displaystyle \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\geq \left| |2x - 2| - |1| \right| \textrm{ by the Reverse Triangle Inequality} \\ &= \left| 2|x - 1| - 1 \right| \\ &> \left| 2(0) - 1 \right| \textrm{ since } |x - 1| > 0 \\ &= |-1| \\ &= 1 \end{align*}$

So that means we have $\displaystyle \displaystyle \begin{align*} 1 < |2x - 3| < 5 \end{align*}$.

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?

This question and the resulting discussion need a comment on the logic of the question.

In my view reply #4 is correct.

Here is the reason. This is a simple implication, *an if-then question*.

If $\displaystyle 0<|x-1|<2$ then what are the possible values of $\displaystyle |2 x-3|~?$

If $\displaystyle P$ is true then $\displaystyle Q$ must be true.

Note that means that if $\displaystyle 0<|x-1|<2$ then $\displaystyle |2x-3|<5$.

If $\displaystyle a=1.5$ then $\displaystyle 0<|a-1|<2$ as well as $\displaystyle |2a-3|=0<5$

Note that if $\displaystyle 0<|x-1|<2$ it is possible that $\displaystyle |2x-3|<1$, example $\displaystyle x=1.8$.

It is also worth noting that if $\displaystyle x=1$ then $\displaystyle 0<|x-1|<2$ is a false statement.

BUT that means that if $\displaystyle 0<|x-1|<2$ then $\displaystyle |2x-3|<5$ is still a true statement.

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: $\displaystyle \displaystyle \begin{align*} |a + b| \leq |a| + |b| \end{align*}$

So $\displaystyle \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\leq |2x - 2| + |-1| \\ &= 2|x - 1| + 1 \\ &< 2(2) + 1 \textrm{ since } |x - 1| < 2 \\ &= 5 \end{align*}$

So we can say $\displaystyle \displaystyle \begin{align*} |2x - 3| < 5 \end{align*}$.

Also, the reverse triangle inequality is useful here to get a lower bound. $\displaystyle \displaystyle \begin{align*} \left| |a| - |b| \right| \leq |a - b| \end{align*}$. So that means

$\displaystyle \displaystyle \begin{align*} |2x - 3| &= |2x - 2 - 1| \\ &\geq \left| |2x - 2| - |1| \right| \textrm{ by the Reverse Triangle Inequality} \\ &= \left| 2|x - 1| - 1 \right| \\ &> \left| 2(0) - 1 \right| \textrm{ since } |x - 1| > 0 \\ &= |-1| \\ &= 1 \end{align*}$

So that means we have $\displaystyle \displaystyle \begin{align*} 1 < |2x - 3| < 5 \end{align*}$.

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 $\displaystyle |2x-3|<5$.

Look at this: $\displaystyle 0<|1.5-1|<2$ and $\displaystyle 0=|2(1.5)-3|<5$.