Question 1142652
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For your attention :


<pre>
    THERE ARE NO functions f, h, r, and l in this post.


    There is ONLY function g(x), and for it, the solution is as follows.
</pre>


<U>Solution</U>


<pre>
The function g(x) is defined as


    g(x) = {{{sqrt((x-2)^2)}}} - {{{x/2}}},


and the equation  g(x) = o  is   


    {{{sqrt((x-2)^2)}}} - {{{x/2}}} = 0.      (1)


The domain is the set of all real numbers.


In this domain, the given equation is equivalent to


    {{{sqrt((x-2)^2)}}} = {{{x/2}}},


which, in turn, is equivalent to


    | x - 2 | = {{{x/2}}}.      (2)


Vertical lines denote the absolute value.


To solve the equation (2), we consider two cases.


<U>Case 1</U>.  x >= 2.   In this case

         | x - 2 | = x-2;  therefore, equation (2) takes the form
    
           x - 2   = {{{x/2}}}.

     It is simplified and solved in this way

           2x - 4 = x

            x = 4.



<U>Case 2</U>.  x < 2.   In this case

         | x - 2 | = -(x-2) = -x + 2;  therefore, equation (2) takes the form
    
           -x + 2   = {{{x/2}}}.

     It is simplified and solved in this way

           -2x + 4 = x

            4 = 3x

            x = {{{4/3}}}



<U>ANSWER</U>.  The given equation has 2 (two, TWO) solutions :  x= 4  and  x= {{{4/3}}}.


         You can check it on your own by substituting these values of x into the original equation.
</pre>


Solved, answered, explained, checked and completed.