1.
express by:
solution set
interval notation
x + 3 / x - 4 0
Solution
The expression is greater than zero when it is positive.
It is positive when the numerator and the denominator have the same sign:
x+3 > 0 and x-4 > 0
or
x+3 < 0 and x-4 < 0
so:
x > -3 and x > 4
for both of these to happen: x > 4
or
x < -3 and x < 4
for both of these to happen: x < -3
Lastly, we must consider when the expression equals zero.
Or, equivalently, whether to include the boundaries (i.e x=4 and x=-3)
the expression equals zero when the numerator equals zero:
x+3 = 0
x = -3
we include this in the set because the expression can be zero.
when x = 4:
the expression becomes infinite (and undefined) and so we exclude this point.
x -3 and x > 4
In interval notation:

2.
(-,-3] U (4,)