solve the inequality and give solution in interval notation?

Question

x/2x+4>=1

Answer 1

I'm going to assume you meant x/(2x+4) >= 1.
Now, when you multiply by 2x+4, it may flip the inequality if 2x+4 is negative. So split it into two cases:
If 2x+4 > 0 (which happens when x>-2), then x>=2x+4, so x+4 <= 0, so x<=-4. Therefore there are no solutions (you can't have x>-2 and x<=-4 at the same time).

If 2x+4 < 0 (which happens when x<-2), then x<=2x+4, so x<=4.
Thus any x < -2 works.

So (-∞,-2).

Answer 2

x/2x+4>=1
multiply both sides by 2x
x+4>=2x
switch the sign
x+4<=2x
get all variables on one side
4<=x