Question

How do you solve for x: (8x)/(x-8)≥12

Answer 1

multiply both sides by (x-8)
8 x ≥ 12 x - 96
96 ≥ 4x
24 ≥ x
============
check with x = 10
80/(2) ≥ 12
40/2 = 20 which is >12 ok

Answer 2

If x > 8, the above solution works, since x-8 > 0 and multiplying does not change the direction of the inequality.

But, what if x=7? 8*8/(7-8) = -56 which is not ≥ 12

The problem is, when x < 8, x-8 is negative, so you need to do
8 x ≤ 12 x - 96
96 ≤ 4x
24 ≤ x
But there is no x which fits that solution.
And of course, x=8 cannot be a solution. why not?

So the only solution set is 8 < x ≤ 24