Question

The line with equation `y=mx+6` has a slope, m, such that `m ∈ [-2,12]`. Use an interval to describe the possible x-intercepts of the line? Please explain in detail how to get the answer.

Answer 1

`[-1/2, 3]`

Explanation:

Consider the high and low values of the slope to determine the high and low value of the x-int. Then we can phrase the answer as an interval.

High:
Let `m=12`:
`y=12x+6`
We want `x` when `y=0`, so
`0=12x+6`
`12x=-6`
`x=-1/2`

Low:
Let `m=-2`
Likewise:
`0=-2x+6`
`2x=6`
`x=3`

Therefore the range of x-ints is `-1/2` to `3`, inclusive.

This is formalized in interval notation as:
`[-1/2, 3]`

PS:
Interval notation:
`[x,y]` is all values from `x` to `y` inclusive
`(x,y)` is all values from `x` to `y`, exclusive.
`(x, y]` is all values from `x` to `y` excluding `x`, including `y`
...
"[" means inclusive, "(" means exclusive.
Note: `oo` is always exclusive. so `x>=3` is `[3,oo)`