Question 476572
.
its all about non linear inequalities. (x-5)(x-4) is greater than or equal to 0.
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        The methods Theo uses to teach students in his posts very often 

        are monstrously verbose, and needlessly long.


        The true Math explanations are usually much shorter, much more straightforward, 

        much clearer, and more instructive.



<pre>
So, they want you solve inequality  (x-5)*(x-4) >= 0.


In this problem the function is presented as the product of two linear binomials, (x-5) and (x-4).


In order for the product of these two binomials would be non-negative, the factors/binomials 
should be of the same sign or zero.



First factor,  (x-5) is negative or zero in the area x <= 5 and non-negative or zero in the area x >= 5.

Second factor, (x-4) is negative or zero in the area x <= 4 and non-negative or zero in the area x >= 4.



Thus we see that both factors are negative or zero in the domain  x <= 4,

             and both factors are non-negative in the domain  x >= 5.


Thus the solution set is the union of two semi-infinite intervals  x <= 4 and x >= 5.


At this point, the solution is fully complete.


<U>ANSWER</U>.  The solution is the set  x <= 4  or  x >= 5.

         i.e. the area out of the interval between the roots of the function (but the solution set includes the roots).


To make sure that the solution is correct, you may plot the function.  See this plot below.


    {{{graph(400,300, -2, 9, -2, 6,
              (x-5)*(x-4))
}}}


               Plot y = (x-5)*(x-4)



The function is a parabola opened upward. Its branches are in the upper (non-negative) half-plane out the interval
on x-axis between x-intercepts, but including the x-intercepts.


The part of the parabola between the roots is in the lover (negative) half-plane.


So, this figure confirms the logical reasoning in algebraic solution.
</pre>

By the way, the answer in the post by @Theo, &nbsp;which is given as  &nbsp;&nbsp;x < 4, &nbsp;x > 5,  
is &nbsp;INCORRECT: &nbsp;it excludes the roots of the function.


Actually, &nbsp;&nbsp;the roots must be included.



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