Solving seemingly easy equations with absolute value

[Darya]

Hi! Today I realized I cannot actually solve equations with absolute value correctly.
I have an equation $$| \frac{x}{x+y} | \leq 1$$My question is what's the best strategy to solve these and what did I forget in my solution:
$$-1 \leq \frac{x}{x+y} \leq 1$$
$$-x-y \leq x \leq x+y$$$$y \geq 0, y \geq 2x$$
When I draw it, it's not symmetric, although it should be. Any ideas?

 

[Harry_the_cat]

To get your second last line, you need to assume that x+y is positive, so that when you multiply everything by x+y, the inequalities stay the same.
However, if x+y is negative, you will need to reverse the inequalities.
You need to consider both of these cases separately.

 

[Steven G]

How did you get y>2x?
What does y > 0, y>2x mean? Does it mean y > 0 AND y>2x or does it mean y > 0 OR y>2x or are both of these the same?

 

[Darya]

How did you get y>2x?
What does y > 0, y>2x mean? Does it mean y > 0 AND y>2x or does it mean y > 0 OR y>2x or are both of these the same?

I think its y<=-2x in the second inequality
I just looked at the -x-y<=x<=x+y by parts ?Anyways, I already understood my mistake and have a correct solution

 

[Darya]

To get your second last line, you need to assume that x+y is positive, so that when you multiply everything by x+y, the inequalities stay the same.
However, if x+y is negative, you will need to reverse the inequalities.
You need to consider both of these cases separately.

thank you so much!!

 

[Steven G]

I think its y<=-2x in the second inequality
I just looked at the -x-y<=x<=x+y by parts ?Anyways, I already understood my mistake and have a correct solution

Nope, it is not y<=-2x. Try again.

 

[Darya]

y

Nope, it is not y<=-2x. Try again.

>=-2x I meant ? Is it okay?