What is the solution set of {x | x < -3} ∩ {x | x > 5}?

Question

1.) all numbers except -3 and 5

2.) the numbers between -3 and 5

3.) the empty set

4.) all real numbers

Answer 1

∩ means "intersect."  It means where the thing before it and the thing after it are true at the same time.

No number is less than -3 and greater than 5 at the same time, so the answer is 3) (the empty set)

The empty set is sort of the answer when there is no answer.

For example, if we want to solve the system of equations x + y = 7, 2x + 2y = 8 we get:

2x + 2y = 8

2(x+y) = 8

x+y = 4

but we also have x+y=7, which leads to:

7 = 4

This can never happen, so there's no combination of values for x and y that make x + y = 7, 2x + 2y = 8 true at the same time.  The answer is the empty set.