How do you solve #|x-4|>3#?

2 Answers
Oct 17, 2017

#color(blue)(x > 7)# or #color(green)(x< 1)#

Explanation:

Remember this:
if #abs(a) < 3#
then #color(blue)(a < 3)# and #color(green)(a > -3)#

if #abs(b) > 3#
then #color(blue)(b > 3)# or #color(green)(b <-3)#

If that doesn't make sense, draw it out on a number line to convince yourself that it does.

Using that rule we can do this to our inequality:

#abs(x-4) > 3#
#color(blue)(x-4 > 3)#
#color(green)(x-4 < -3)#

#color(blue)(x-4 > 3)#
#color(blue)(x > 7)#

#color(green)(x-4 < -3)#
#color(green)(x< 1)#

#color(blue)(x > 7)# or #color(green)(x< 1)#

Oct 17, 2017

#x>7 or x<1#

Explanation:

#|x-4|>3#
Solve the absolute value
We know
either #x -4 >3 or x -4<-3#

We have two possibilities. The first one is #x - 4>3# and the second one is #x -4<-3#
Solve the first one
#x - 4>3#
Add #color(red)(4)# to both sides
#x cancel(-4) cancelcolor(red)(+4)>3 color(red)(+4)#
#x >7#

Solve the second possibility
#x - 4 <-3#
Add #color(green)(4)# to both sides
#x cancel(-4) cancelcolor(green)(+4) < -3 color(green)(+4)#
#x <1#