Question 1116168
.
<pre>
{{{2/x}}} <= {{{x/2}}}      (1)

{{{2/x}}} - {{{x/2}}} <= {{{0}}}

{{{4/(2x)}}} - {{{x^2/(2x)}}} <= {{{0}}}   (written with the common denominator 2x )

{{{(4-x^2)/(2x)}}} <= {{{0}}}

{{{((2-x)(2+x))/(2x)}}} <= {{{0)}}}     (2)


On the left side, we have a rational function of the three factors (2-x), (2+x) and (2x).


There are three critical values, where the factor become equal to zero and change their sign: -2, 0, and 2.


And there are four interval to analyse:  a) x < -2;  b) -2 <= x < 0;  c) 0 < x < 2;  and  d) x >= 2.


In the interval  a) x < - 2  factors (x+2) and (2x) are negative;  factor (2-x) is positive.  So inequality (2) is FALSE.


In the interval  b) -2 <= x < 0  the factor (x+2) is positive, factor (2x) is negative; factor (2-x) is positive.  
So, the whole function in the left side of (2) is negative, and the inequality (2) IS TRUE.


In the interval  c) 0 < x < 2  the factors (x+2)  and (2x)  are positive;  the factor (x-2) is positive; so inequality (2) is FALSE.


In the interval  d) x >= 2  factors (x+2) and (2x) are positive;  factor (2-x) is negative.  So, the whole function in the left side of (2) is negative, and the inequality (2) IS TRUE.


Thus the inequality (2) is TRUE in these two intervals

     -2 <= x < 0    and    x >= 2.


Since inequalities (1) and (2) are EQUIVALENT (!),  your answer is:


<U>Answer</U>.  The original inequality is true  at  -2 <= x < 0    and    x >= 2.

         The solution is the set  [{{{-2}}},{{{0}}})  U  [{{{2}}},{{{infinity}}}].
</pre>

Solved.


------------
To see many other similar solved problems for inequalities for rational functions, &nbsp;look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Inequalities/Solving-inequalities-for-rat-functions-with-num-and-denom-factored-into-a-product-of-linear-binomials.lesson>Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials</A> 

in this site.


Also, &nbsp;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 lesson is the part of this online textbook under the topic &nbsp;"<U>Inequalities</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.