Question

Solve the compound inequality below. Describe your steps.
l x-3 l < 4 and l x+2 l >8

Answer 1

The compound inequality \|x-3|<4\ and \|x+2|>8\.
To solve an absolute value inequality, isolate the absolute value expression on one side of the inequality If \|x|<a\, then \-a<x<a\ If \|x|>a\, then \x<-a\ or \x>a\ The solution to a compound inequality with "and" is the intersection of the solutions to each individual inequality.

To solve an absolute value inequality, isolate the absolute value expression on one side of the inequality If \|x|<a\, then \-a<x<a\ If \|x|>a\, then \x<-a\ or \x>a\ The solution to a compound inequality with "and" is the intersection of the solutions to each individual inequality
Step 1 …Solve the first inequality |x-3|<4.|x-3|<4 -4<x-3<4 -4+3<x<4+3

Step 2 …Solve the second inequality |x+2|>8.|x+2|>8 x+2<-8 or x+2>8 x<-10 or x>6

Step 3…Find the intersection of the solutions to each inequality.
Solution
The solution to the compound inequality is 6<x<7