Question 1085663
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<pre>
The critical points are 6-2x = 0, i.e. x = 3;  and x-1=0, i.e. x = 1.


If x < 1, then        6 - 2x > 0 and hence |6-2x} = (6-2x);

                      x - 1 < 0  and hence |x-1| = (1-x);

                      thus the entire function is f(x) = (6-2x) + (1-x) - 2x = 7 - 5x. 



If 1 < x < 3,    then 6 - 2x > 0 and hence |6-2x} = (6-2x);

                      x - 1 > 0  and hence |x-1| = (x-1);

                      thus the entire function is f(x) = (6-2x) + (x-1) - 2x = 5 - 3x.



Lastly, if x > 3 then 6 - 2x < 0 and hence |6-2x} = (-6+2x);

                      x - 1 > 0  and hence |x-1| = (x-1);

                      thus the entire function is f(x) = (-6+2x) + (x-1) - 2x = -7 + x.


So you have the expressions for f(x) piecewise linear and can draw it like this


{{{graph( 330, 330, -5.5, 10.5, -5.5, 10.5,
          abs(6-2x)+abs(x-1)-2x
)}}}


Plot y = |6-2x|+|X-1|-2X
</pre>


On plotting absolute value functions see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-plot-functions-containing-Linear-Terms-under-Abs-Value-sign-L-1.lesson>How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 1</A> 

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-plot-functions-containing-Linear-Terms-under-Abs-Value-sign-L-2.lesson>How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 2</A> 

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/absolute-value/HOW-TO-plot-functions-containing-Linear-Terms-under-Abs-Value-sign-L-3.lesson>How to plot functions containing Linear Terms under the Absolute Value sign. Lesson 3</A> 

in this site.



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic
"<U>Plotting Absolute values functions </U>".



The strategy is to break up the entire set of real numbers into sub-domains (ranges) where the absolute value of linear term 
is a linear function, &nbsp;and then to plot all these piece-wise linear functions.