Question

identify the solution set of l x + 3 l + 7 < 10 and describe the graph in a complete sentence. I'm really stuck on this question. I would really appreciate the help.

Answer 1

When dealing with absolute values it is helpful to simply split the equation into two different cases. One case where the term in the absolute value is positive, and another case where it is negative. Then you have two regular equations and can solve them using regular methods.

Answer 2

Does that make sense? So in this problem, first we can subtract 7 from both sides to get \(|x+3|<3\), then we just split it up into two cases. In one case, \(x+3>0\), so \(|x+3|=x+3\), so we have \(x+3<3\implies x<0.\) In the second case, \(x+3<0\), so \(|x+3|=-(x+3)\), so we have \(-(x+3)<3\implies -x-3<3\implies -x<6\implies x>-6.\) So now we have two solutions, \(x<0\) and \(x>-6.\) We can write these together in a compound expression by saying \(-6<x<0.\) We could also say that \(x\in(-6,0)\), meaning that x is in the open interval from -6 to 0.

Answer 3

Yeah that actually makes a lot of sense. Thank you so much that helped me a lot ! I know get how to do it. :)