Question

Solve for x under the assumption that x>0. Enter your answer in interval notion using grouping symbols.
X-48/x<-8

Answer 1

[x-48]/x<-8
x-48<-8x
9x<48
x<48/9

Answer 2

(x-48)/x < -8
You cannot just multiply by x as shown above, because x might be negative, which would then reverse the direction of the inequality.

If x > 0, then
x-48 < -8x
9x < 48
x < 48/9
so, 0 < x < 48/9

If x < 0, then
x-48 > -8x
9x > 48
x > 48/9
But that's out, since we assumed x<0.

If you plot the curve, you will see that a simple "x < 48/9" is not quite true.

Answer 3

OR

(x-48)/x < -8
(x-48)/x + 8 < 0
(x-48 + 8x)/x < 0
(9x - 48)/x < 0
but we are told that x > 0, then
9x-48 < 0
9x < 48

x < 48/9