Question 1120102
.
First, the formula is ambiguous, since it can be read in different ways.


One way to read it is  f(x) = {{{(2x-3)/((x-2)*(x-4))}}}.


Another way is  f(x) = {{{((2x-3)/(x-2))*(x-4))}}}.


To make it UNAMBIGOUS, you must use parentheses, showing which part is the numerator, and which is the denominator.



I will read the formula f(x) = {{{(2x-3)/((x-2)*(x-4))}}}.



<pre>
The critical points are  x= {{{3/2}}}    (the zero of the numerator and the zero of the function),

                         x= 2  and  x= 4    (the zeroes of the denominator).


They divide the number line in four intervals 

     {{{-infinity}}} < x < {{{3/2}}},    {{{3/2}}} < x < 2,    2 < x < 4    and    4 < x < {{{infinity}}}.


1.  In the interval  {{{-infinity}}} < x < {{{3/2}}}  all three factors (2x-3), (x-2) and (x-4) are negative.  

    So, the function f(x) is negative as the product/quotient of three negative numbers.

    So, this interval  {{{-infinity}}} < x < {{{3/2}}}  is the part of the solution domain.



2.  In the interval  {{{3/2}}} < x < 2,  the factor (2x-3) is positive, while (x-2) and (x-4) are negative. 

    So, the function f(x) is positive as the product/quotient of one positive and two negative numbers.

    So, this interval  {{{3/2}}} < x < 2,  is not the part of the solution domain.



3.  In the interval  2 < x < 4,  the factors (2x-3) and  (x-2) are positive, while  (x-4)  is negative. 

    So, the function f(x) is negative as the product/quotient of two positive and one negative numbers.

    So, this interval  2 < x < 4,  is the part of the solution domain.



4.  In the interval  4 < x,  the factors (2x-3),  (x-2)  and  (x-4)  are positive. 

    So, the function f(x) is positive as the product/quotient of three positive numbers.

    So, this interval  4 < x  is not a part of the solution domain.


<U>Answer</U>.  The solution set is the union of two intervals  ({{{-infinity}}},{{{3/2}}}) U (2,4).
</pre>

Solved.


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