Determine solution set of the inequality

$$|x-1|\geq 3|x+1|$$

Question

Determine solution set of the inequality 

$$|x-1|\geq 3|x+1|$$

anonymous · Answer

i think you need to work in cases 
if $x>1$ then $|x-1|=x-1$ and $|x+1|=x+1$
first job it to solve 
$$x-1\geq 3(x+1)$$

anonymous · Answer

if $x<-1$ then $|x-1=1-x$ and $|x+1|=-x-1$
etc

anonymous · Answer

the solution is -2<x<-1/2 but i don't know how we found it..

anonymous · Answer

ok lets go slow

anonymous · Answer

ok..

anonymous · Answer

if $x>1$ then both $x-1$ and $x+1$ are positive, right?

anonymous · Answer

yes

anonymous · Answer

which means they are equal to their absolute values
in other words $|x+1|=x+1$ and $|x-1|=x-1$

anonymous · Answer

so if $x>1$ you need to solve 
$$x-1\geq 3(x+1)$$

anonymous · Answer

we solve in a couple of steps $$x-1\geq 3x+3$$$$-1\geq 2x+3 \-4\geq 2x \x<-2$$ but here we have a problem because we were assuming that $x>1$ but came up with $x<-2$ so if $x>1$ there is NO solution

anonymous · Answer

now we try again with 
$$x<-1$$ but this time both $x-1$ and $x+1$ are negative, and so 
$$|x+1|=-x-1$$ and 
$$|x-1|=-x+1$$

anonymous · Answer

we solve 
$$-x+1\geq 3(-x-1)$$ in the same number of steps 
$$-x+1\geq -3x-3$$
$$2x+1\geq -3$$
$$2x\geq -4$$
$$x\geq -2$$

anonymous · Answer

that means if $x<-1$ then also $x>-2$ so we know we are good from $-2$ up to $-1$

anonymous · Answer

thanks Satellite, what about 1/2 we still didn't get it..

anonymous · Answer

(sorry i got internet problem before)