Question

Determine all intervals on which f(x) < 0.

Answer 1

Answer:
`(-\infty, -7) \cup (-3, 0) \cup (4, 8)`

=============================================

Step-by-step explanation:

Recall that y = f(x)

Asking when f(x) < 0 is the same as y < 0.

We focus on the portions of the curve below the x axis where y is negative.

The interval to the left of x = -7 is the first such portion below the x axis.

This gives
`x < -7` aka
`-\infty < x < -7` and that condenses to the interval notation of
`(-\infty, -7)`. We use a curved parenthesis to exclude x = -7. This is because x = -7 is a root and means y = f(-7) = 0.

The next interval below the x axis is when
`-3 < x < 0` which condenses to the interval notation
`(-3,0)`

The last portion below the x axis is when
`4 < x < 8` which condenses to the interval notation
`(4,8)`

--------

In summary we found these three separate regions below the x axis

`(-\infty, -7)` and
`(-3, 0)` and
`(4,8)`

Each refers to interval notation.

Glue the regions together with the union symbol and we arrive at the final answer
`(-\infty, -7) \cup (-3, 0) \cup (4, 8)`