Question

how to find the solution to -4x ≥ 16

Answer 1

if we wrote
16 > 4 (which is true)
and divided by 4, we would get
4 > 1 (still true)

if we write
-16 < -4 (which is true)
and divided by -4
4 < 1 (not true!)

dividing or multiplying by a negative *swaps* the relation

to stay out of trouble we could always add or subtract:

-4x ≥ 16

add +4x to both sides
-4x + 4x ≥ 16 +4x
0 ≥ 16 +4x
now add -16 to both sides
-16 ≥ 4x
now we can divide by positive 4 (which is always safe) to get
-4 ≥ x

of course, we could memorize the rule
if you divide by a negative number, swap > to < (or vice versa)