When we join two inequalities with an “or,” we unite their graphs – we include all solutions to *both* inequalities. Let’s first look at graphs of each of the four inequalities first, and then we’ll see when we unite them.
r < 5 starts with a hollow circular mark on 5 (since 5 isn’t included in r), and includes everything from there to negative infinity
r > 5 starts with that same mark, but its solutions include everything from there to positive infinity (except 5)
r > -1 starts at -1 and includes everything from there to positive infinity (except -1), and r < -1 stretches to negative infinity (except -1).
We want to join together two inequalities so that their solutions span from negative to positive infinity (all real numbers); which of our options include negative infinity, and which include positive infinity?
Negative infinity: r < 5, r < -1
Positive infinity: r > 5, r > -1
We could try joining r > 5 and r < -1, but then we’d be exclude all of the real numbers between and including -1 and 5. The graphs of r < 5 and r > -1 pass right by each other and continue infinitely off to negative and positive infinity respectively, so all possible real numbers will be included in their combined range.
So, our solution is “r < 5 or r > -1”.
