|c-2|>6
|3-(x-1)|<8
Rules for getting rid of absolute values:
If A is a POSITIVE number:
|EXPRESSION| < A becomes this---> -A < EXPRESSION < A
|EXPRESSION| > A becomes this---> EXPRESSION < -A OR EXPRESSION > A
So:
|c-2| > 6 becomes this---> c-2 < -6 OR c-2 > 6
add 2 to both sides in each: c < -4 OR c > 8
so the graph of the solution is this number line:
<=======o-----------------------------------o======>
-7 -6 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
And the interval notation for that is:
(-oo, -4) U (8, oo)
*****************************************************
*****************************************************
|3-(x-1)| < 8 becomes this: -8 < 3-(x-1) < 8
remove parentheses: -8 < 3-x+1 < 8
combine 3 and +1 : -8 < 2-x < 8
subtract 2 from all 3 sides: -10 < -x < 6
divide thru by -1 which
reverses the < to > : 10 > x > 6
that is now greatest to smallest
which can be written smallest
to greatest: 6 < x < 10
The graph is on this number line:
----------------------o===========o------------
-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
which in interval notation is (6, 10)
Note: (6, 10) looks like a point on an xy-axis,
but it isn't. We have to go by context to determine
whether (6, 10) represents an interval notation or
a point like (x, y). Here it is interval notation.
Edwin
