How do you solve the inequality #2abs(7-x)+4>1# and write your answer in interval notation?

2 Answers
Jul 23, 2018

Answer:

#[11/2,17/2]#

Explanation:

Isolate the absolute value (in red):
#2\color{red}{|7-x|}+4\gt1#
#2\color{red}{|7-x|}\gt1-4#
#\color{red}{|7-x|}\gt\frac{-3}{2}#


Expand absolute value:
#-3/2\lt7-x\lt-(-3/2)#
#-3/2\lt7-x\lt3/2#
#-3/2-7\lt-x\lt3/2-7#

Note the sign change here. This happens when you divide by negatives.
#3/2+7\color{tomato}{\gt}x# and #x\color{tomato}{\gt}-3/2+7#
Effectively, this means #-3/2+7\ltx\lt3/2+7#

Simplifying new inequality:
#-3/2+7\ltx\lt3/2+7#
#-3/2+14/2\ltx\lt3/2+14/2#
#11/2\ltx\lt17/2#


In interval notation:
#[11/2,17/2]# (brackets are used when you have #\lt,\gt# signs).

Jul 23, 2018

Answer:

The solution is #x in (-oo,+oo)#

Explanation:

Let

#7-x=0#

#=>#, #x=7#

#AA x in RR#

#2|7-x|>0#

And

#2|7-x|+4>1#

The solution is

#x in (-oo,+oo)#

graph{2|7-x|+3 [-21.99, 29.33, -11.31, 14.36]}