Question

What is the rule for when we reverse the inequality?

If anyone can answer that, they get a medal!

Answer 1

if you have
-x < 1

add +x to both sides
+x-x< 1+x or 0 < 1 + x
add -1 to both sides
-1 < 1 - 1 +x or
-1 < x

so we see
-x < 1 and -1 < x mean the same thing

Answer 2

which can also be written
-x < 1 is the same as
x > -1