Question 100079
Montrose and Salem are 10 miles apart. Karen lives 12 miles from Salem.  What are the possible distances Karen lives from Montrose?
<pre><font size = 4 color = "darkblue"><b> 
Let the point M be Montrose and the point S be Salem

{{{drawing(400,375,-4,24,-12,12,
line(0,0,10,0), locate(4,-.2,"10_mi"),
locate(-.2,.6,o), locate(9.8,.6,o), locate(0,1.5,M), locate(10,1.5,S)   
 )}}}

Since Karen lives 12 miles from S, we'll draw a circle with center S 
and radius 12 miles:

{{{drawing(400,375,-4,24,-12,12,
circle(10,0,12), line(0,0,10,0), locate(4,-.2,"10_mi"),
locate(-.2,.6,o), locate(9.8,.6,o), locate(0,1.5,M), locate(10,1.5,S)   
 )}}}

Karen lives somewhere on that 12 mile radius circle. Let's draw in the 
complete horizontal diameter of that circle, and label its endpoints A
and B:

{{{drawing(400,375,-4,24,-12,12, locate(15,-.2,"12_mi"),locate(-3,0,A),locate(23,0,B), 
circle(10,0,12), line(-1.7,0,22.1,0), locate(4,-.2,"10_mi"), locate(-1,-.2,"2"),
locate(-.2,.6,o), locate(9.8,.6,o), locate(0,1.5,M), locate(10,1.5,S), locate(-1,-1.1,"mi")   
 )}}}

The closest Karen could live to M is 2 miles if she lives at A.
The farthest Karen could live from M is 22 miles if she lives at B.

Therefore she lives 2 miles or more from Montrose and 22 miles or 
less from Montrose.  So if her home's distance from Melrose is 
represented by the letter D, then the inequality is

      2 <u><</u> D <u><</u> 22

Edwin</pre>