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.

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Feb 22, 2018

Answer:

#[-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)#

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