How do you solve and write the following in interval notation: #x − 3 ≤ 2# AND #-x / 3 < 2#?

1 Answer
Jan 21, 2017

Answer:

#x-3<=2# and #-x/3<2# #<=># #x in (-6,5]#

Explanation:

#x-3<=2# and #-x/3<2#

Then

for #x-3<=2#

#<=>#

#x<=2+3=5# Add three to both sides

Therefore

#x<=5 <=> x in (-oo,5]#

This means #x# is in the set of points between #-oo# to and including #5#. So, #x# could be #5# or any number less than #5#.

likewise

for #-x/3<2#

#-x<2(3)=6# multiply both sides by 3

Then

#-x<6# divide both sides by -1

#<=> x > -6# then # x in (-6,oo)#

This means #x# is in the set of points greater than #-6# but not including #-6#. Then #x# can be any number greater than #-6# but not #-6#.

If #x<=5# and #x> -6#, then #x in (-oo,5]nn(-6,oo)#. Where the upside down cup #nn# means "intersection", meaning #x# is in both the first and second set. Then #x in (-6,5]# because #x# cannot be less than or equal #-6#, and it is also less than any number greater than #5#.