# Absolute value question

#### geekay

##### New member
If 0<a<2b applies, then

|a-2b| = 2b-a

Could someone please explain me why the above is true?

Sorry if this is posted in the wrong section.

Much appreciated.

#### Cubist

##### Full Member
Is the following true? And if so, then can you simplify the RHS?

|a-2b| = |-(a-2b)|

#### geekay

##### New member
Thanks for the reply.

if I open the parentheses on the RHS, it then becomes |-1*(a-2b)|=|-a+2b|, would it not?

Guess this is wrong way of thinking. So should one instead use |-1*(a-2b)|=|-1|*|a-2b|=|a-2b| ?

Perhaps my problem is this: don’t know how to get rid of the absolute value signs.

#### geekay

##### New member
Hey, guess I got it now! Thank you so much!

#### Dr.Peterson

##### Elite Member
If 0<a<2b applies, then

|a-2b| = 2b-a

Could someone please explain me why the above is true?

Sorry if this is posted in the wrong section.

Much appreciated.
Recall the piecewise definition of absolute value: if x>=0, then |x| = x; if x<0, then |x| = -x.

Which case applies here?

#### JeffM

##### Elite Member
$$\displaystyle \text {Definition: } c < 0 \implies |\ c \ | = - c;\ c \ge 0 \implies |\ c \ | = c.$$

Good so far?

$$\displaystyle \text {Let } d \text { be any real number.}$$

$$\displaystyle \therefore d < 0, d = 0, \text { or else } d > 0.$$

$$\displaystyle \text {CASE I: } d < 0 \implies \\ |\ d \ | = - d > 0 \implies |\ - d \ | = - d \implies |\ d \ | = |\ - d \ |.$$

$$\displaystyle \text {CASE II: } d = 0 \implies \\ |\ d \ | = 0 = - 0 \implies |\ - d \ | = |\ - 0 \ | = |\ 0 \ | = 0 \implies |\ d \ | = |\ - d \ |.$$

$$\displaystyle \text {CASE III: } d > 0 \implies \\ |\ d \ | = d > 0 \implies - d < 0 \implies |\ - d \ | = - (-d) = d \implies |\ d \ | = |\ - d \ |.$$
$$\displaystyle \therefore \text {in all cases, } |\ d \ | = |\ - d \ |.$$

Any questions?

Now apply that theorem to your problem.

#### geekay

##### New member
Thanks again for the help, guys. Great community!