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

1 Answer
Jun 2, 2017

Answer:

#-3 < x <= 2# OR #6<=x<11#

Explanation:

First, recall the rules of absolute values in inequality expressions.

Absolute value is less than something rule
If #abs(A) < B#, that's the same as saying #-B < A < B#

Absolute value is greater than something rule
If #B<=abs(A)#, that's the same as saying either #B<=A# or #A<=-B#

The inequality we are given can be broken into two parts.

First part: Less than rule
#abs(4-x)<7#

Applying the "less than rule" gives
#-7<4-x<7#

Subtract #4# from all three sides
#-11 < -x < 3#

Divide by #-1# and flip the inequality signs
#-3 < x < 11#

Second part: Greater than rule
#2<=abs(4-x)#

Applying the "greater than rule" gives
Either #2<=4-x# or #4-x<=-2#

Solving both for #x# gives

Either #x<=2# or #x>=6#

Finally, combine all solutions into one

The solution: #-3 < x < 11# gives the most-negative and most-positive outside boundaries.

The solution: Either #x<=2# or #x>=6# gives the inner boundaries.

ANSWER: #-3 < x <= 2# OR #6<=x<11#

Here is what the graph would look like:
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