How do you solve #|t + 4| > 10#?

Question

2488 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-03T03:41:34

#tin(-∞,-14)#U#(6,∞)#

Using the definition of the absolute value, we are able to split this inequality with absolute values into two without absolute values:

#t+4>10#, when #t+4>=0#
#-(t+4)>10#, when #t+4<0#

When solving each inequality separately, we get two conditions for t:

#t+4>10#
#(t+4)-4>10-4#
#t>6#

#-(t+4)>10#
#-t-4>10#
#(-t-4)+4>10+4#
#-t>14#
#-(-t)> -(14)#
#t<-14#

We must then intersect each condition with its corresponding restriction:

#(t>6)# ∩ #(t+4>=0)#
#(t>6)# ∩ #(t>=-4)#
#t>6#

#(t<-14)# ∩ #(t+4<10)#
#(t<-14)# ∩ #(t<-4)#
#t<-14#

Lastly, we must union these two new conditions:

#(t>6)# U #(t<-14)#
#tin(-∞,-14)#U#(6,∞)#

Answer 2 · 2018-03-03T03:43:02

Read below.

If you have #absx>b#, then you can set it up as:

#x>b# or #x<-b#

Therefore,

#abs(t+4)>10# becomes #t+4>10# or #t+4<-10#

We solve each inequality.

#t+4>10#

#t>6#

#t+4<-10#

#t<-14#

#t# either has to be greater than #6# or less than #-14#.