Question 156751
Graph each Absolute Value equation by writting two linear equations.

{{{y= abs(3x + 6)}}}
<pre><font size = 5 color = "indigo"><b>
One linear equation is gotten from the case 
when what is between the absolute value bars,
3x+6, is negative, that is, less than 0.

The other linear equation is gotten from the 
case when what is between the absolute value 
bars, 3x+6, is either positive or zero, that 
is, greater than or equal to 0.

The first linear equation is gotten when 3x+6
is negative, but if 3x+6 is negative, then if 
we multiply 3x+6 by -1, then it will become 
positive, and so -1(3x+6) will be positive, 
and that will be the absolute value of 3x+6,
when 3x+6 is negative.  So the first linear 
equation is

{{{y=-1(3x+6)}}} or 

{{{y=-3x-6}}}.

So we draw the graph of that line:

{{{drawing(400,400,-8,4,-8,4,
   graph(400,400,-8,4,-8,4),
line(-9,21,5,-21)   )}}}

However, since we are requiring that
what is between the absolute value
bars, 3x+6, is less than 0, we must
only use the part of that line where
{{{3x+6 < 0}}}.  So we solve that:

{{{3x+6 <0}}}
{{{3x<-6}}}
{{{x<-2}}}

Therefore we must chop off the line
to the RIGHT of where x is equal to
-2.  So the graph is only this part
of the line:

{{{drawing(400,400,-8,4,-8,4,
   graph(400,400,-8,4,-8,4),
line(-9,21,-2,0)   )}}}

and it DOES NOT include the point
(-2,0).

-----------------

Now The second linear equation is gotten 
when 3x+6 is positive or zero, and if 3x+6 
is positive or zero, then we do not need
abslute value bars at all. That is, when 
3x+6 is positive or zero, the absolute 
value of 3x+6 is simply 3x+6!  So the
second linear equation is just:

{{{y=3x+6}}} 

So we draw the graph of that line on
the same set of axes:

{{{drawing(400,400,-8,4,-8,4,
   graph(400,400,-8,4,-8,4),
line(-9,-21,5,21),line(-9,21,-2,0)   )}}}

However, since we are requiring that
what is between the absolute value
bars, 3x+6, is greater than or equal
to 0, we must only use the part of 
that line where {{{3x+6 >= 0}}}.  So
we solve that:

{{{3x+6 >=0}}}
{{{3x>=-6}}}
{{{x>=-2}}}

Therefore we must chop off that line
to the LEFT of where x is equal to -2.
So the FINAL graph is only this V-shaped
graph:

{{{drawing(400,400,-8,4,-8,4,
   graph(400,400,-8,4,-8,4),
line(-9,21,-2,0),line(5,21,-2,0)   )}}}

and and the part slanting up to the right
DOES include the point (-2,0), 

So the absolute value equation 

{{{y=abs(3x-6)}}}

can be written as the piecewise function 

{{{y=system(matrix(2,3,-3x+6,when,x<-2,3x-6,when,x>=-2))}}}

without using any absolute value bars!

Edwin</pre>