Question

Can anyone tell me how to solve:
|x|+|x+2|=3

Answer 1

Hi! :)
\(\Large\bf \color{#008353}{\text{Welcome to OpenStudy! :)}}\)

Answer 2

I think I can help you with this, gimme couple minutes.
Gotta eat my yogurt -_-

Answer 3

Hi & thank you!
You can take your time :)

Answer 4

Ok ok so do you remember this little tidbit about the absolute function?\[\Large |x|\quad=\quad \cases{x,\qquad x>0\\-x,\quad x<0}\]

Answer 5

Well when we have a shift, we have to shift that point where the sign switches also.\[\Large\bf\sf |x+2|\quad=\quad \cases{\qquad x+2,\qquad x+2>0\\-(x+2),\qquad x+2<0}\]Which leads to,\[\Large\bf\sf |x+2|\quad=\quad \cases{\qquad x+2,\qquad x>-2\\-(x+2),\qquad x<-2}\]

Answer 6

So we'll look for solutions in each region.
For \(\Large\bf\sf x>0\),

\[\Large |x|\quad=\quad \cases{\color{orangered}{x,\qquad x>0}\\-x,\quad x<0}\]\[\Large |x+2|\quad=\quad \cases{\color{orangered}{\qquad x+2,\qquad x>-2}\\-(x+2),\qquad x<-2}\]

Answer 7

Both pieces are positive when x is greater than zero, right?
See what I'm doing kinda? D:

Answer 8

So for x>0 we have,\[\Large\bf\sf |x|+|x+2|=3\]Becomes,\[\Large\bf\sf x+x+2=3\]

Answer 9

I understand so far xD Then we have to find the x<0 too?

Answer 10

We'll need to be a little bit more specific than that.
We actually have 3 regions to look at.

Next we'll look here -2<x<0.
x is between 0 and -2.

Answer 11

When -2<x<0,
\[\Large |x|\quad=\quad \cases{x,\qquad x>0\\\color{orangered}{-x,\quad x<0}}\]
\[\Large |x+2|\quad=\quad \cases{\color{orangered}{\qquad x+2,\qquad x>-2}\\-(x+2),\qquad x<-2}\]

Answer 12

So for -2<x<0 we have,\[\Large\bf\sf |x|+|x+2|=3\]Becomes,\[\Large\bf\sf -x+x+2=3\]

Answer 13

And what can we determine here? :o

Answer 14

Ooh .. So that one doesn't work? I think I understand now.
Then for x<-2 :
-x-x-2=3 ?

Answer 15

No solution in the second area? Correct!
For x<-2? Yes very good!

Answer 16

I understand now. Kinda confused me at first xD; Thank you very much for the detail explanation!! :D

Answer 17

Yah it's kind of a weird problem *|c:/*
np