CAN SOMEONE CHECK THIS.....

Watch your notation
When you subtracted -4x and put it in parenthesis (-4x) it looks like multiplication

I use the following notation for steps
m(6) means multiplying the equation by 6
d(6)-dividing by 6
a(6)-adding 6 to both sides
s(6)-subtracting 6 from both sides
The book seems to be trying to emphasis that when you multiply by -1 you flip the inequality sign

A simpler solution (IMO) is
1/2x+3<2/3x
m(6) 3x+18<4x
s(3x) 18<x
flip x>18 (put in variable first notation)

The book solution
3x+18<4x
s(4x) -x+18<0
s(18) -x<-18
m(-1) x>18 (the inequality sign flips direction when you multiply by negative numbers)

Another solution
1/2x+3<2/3x
s(1/2x) 3<(2/3-1/2)x
3<1/6x
m(6) 18<x
flip x>18 (put in variable first notation)

They're all result in the same answer

Rap

Thanks.
When I got to this part of an inequality 3X + 18 < 4X ---> 3X s-4X + 18 < 4X s-4X
It's actually subtracting 4 from both sides. But I mean can you really subtract 4 from both sides and leave other two numbers by itself on the other side of the equation?
3X + 18 < 4X
-4X <---> -4X
------------------
-X +18 < 0

so therefore is that actually the right way of doing it? I mean the answer is right but, shouldn't you subtract 3X from both sides of the inequality?

Either way is right. In reality I would subtract 3x from both sides resulting in 18<x and then rewrite this as x>18, but the other way (subtracting 4x) will get you to the same place.

Rap

Alright thanks.