| x - 2 | = | 2x - 1 |

[burgerandcheese]

If | x | = a means x = ±a

I know the given solution to | x - 2 | = | 2x - 1 | is x - 2 = ± (2x - 1) or 2x - 1 = ±(x - 2). But I don't understand. We don't know if x - 2 or 2x - 1 is either positive or negative, how come we can solve it like that?

 

[Harry_the_cat]

When would |a| = |b|?

 

[HallsofIvy]

Do both. If x- 2= 2x-1 then, subtracting x from both sides, -2= x- 1. Then adding 1 to both sides, -1= x.

If x- 2= -(2x-1)= -2x+ 1, adding 2x to both sides, 3x- 2= 1. Adding 2 to both sides, 3x= 3. Finally divide both sides by 3: x= 1.

x= 1 and x= -1 satisfy |x- 2|= |2x- 1|.

 

[burgerandcheese]

When would |a| = |b|?

when a and b are the same but either one could be positive or negative

 

[Harry_the_cat]

Yes, in other words a= b or a= -b

 

[pka]

If | x | = a means x = ±a
I know the given solution to | x - 2 | = | 2x - 1 | is x - 2 = ± (2x - 1) or 2x - 1 = ±(x - 2). But I don't understand. We don't know if x - 2 or 2x - 1 is either positive or negative, how come we can solve it like that?

Can you solve this \(\displaystyle (x-2)^2=(2x-1)^2~?\)
If so then compare the solutions.

 

[burgerandcheese]

I get it now! Thanks so much everyone ?