How do you solve the inequality #abs(2x+1)<=6-x# and write your answer in interval notation?

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Answer 1 · 2017-04-03T15:20:42

#-7 \leq x \leq 5/3#

Since #|a|\leq b \Leftrightarrow -b \leq a \leq b#, the inequality #|2x+1|\leq 6-x#becomes

#-(6-x) \leq 2x+1 \leq 6-x#

#\Leftrightarrow x-6 \leq 2x+1 \leq 6-x#

#\Leftrightarrow x-7 \leq 2x \leq 5-x#

Solve the inequality on the left:
#x-7 \leq 2x \Leftrightarrow -7 \leq x#

Solve the inequality on the right:
#2x \leq 5-x \Leftrightarrow 3x \leq 5 \Leftrightarrow x \leq 5/3#

Combine the two intervals: #-7 \leq x \leq 5/3#