The other person who responded or I guess, tried to help is on some other planet, so as most people know by now, and are doing, "Simply IGNORE the RUBBISH he posted!"

Correct answer: . This means that you have 4 INTERVALS to test, which are: 

Testing these INTERVALS will give you the correct solution(s). When done, you'll find that 2 of these 4 INTERVALS provide solutions. 

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
 .

Solve the inequality 24x + 17 > 20 /x

~~~~~~~~~~~~~~~~~~~~~





The "solution" by @josgarithmetic is  WRONG.  It is  WRONG  from the beginning to the end.





His approach is  WRONG.   His method is  WRONG.   His answer is WRONG.





Below find the correct approach,  the correct method,  the correct solution and the correct answer.





Regarding writing by @josgarithmetic,  ignore it and forget about it.





Solution





24x + 17 > 20 /x.     (1)

We divide the solution in two parts.
In the first part we will consider the domain 0 < x < , where x is positive. 
In the second part we consider the domain  < x 0, where x is negative.

From the very beginning notice that the domain of the given inequality is holed number line with the hole at x= 0, 
where the denominator of the right side (1) is zero.


1.  So, consider the domain x > 0.

    Then we can multiply both sides of (1) by x. The inequality sign remains with no change, since we multiply by positive number.

    Then instead of (1), we get an inequality

    24x^2 + 17x - 20 > 0,  which we consider in the domain x > 0.   (2)

    Apply the quadratic formula to get the roots of the equation 24x^2 + 17x - 20 = 0.

    You will get the roots as  =  = ,   or

     =  = 0.625   and   =  = .

    So, our quadratic function (2) is positive to the right from 0.625 and is negative at the interval (0,0.625).

    Thus the solution to the inequality (1) on the domain x > 0 is  the set { x | x > 0.625}.   (3)

    At this point we completed our consideration for the domain x > 0.
    Now switch our attention to the domain x < 0.


2.  Now, consider the domain x < 0.

    Again, we can multiply both sides of (1) by x, but this time we MUST CHANGE THE INEQUALITY SIGN 
    to the opposite one, since we multiply inequality by the negative number.

    Then instead of (1), we get an inequality

    24x^2 + 17x - 20 < 0,  which we consider in the domain x < 0.   (4)

    The roots of the equation 24x^2 + 17x - 20 = 0 are the same as we found in the part 1. They are  and 0.625.

    So, our quadratic function (4) is negative to the left from 0 to , i.e. is negative at the interval (,0).

    Thus the solution to the inequality (1) on the domain x < 0 is  the set { x |  < x < 0}.   (5)


3.  Combining the solutions (3) and (5), we get the solution to our original inequality (1).

    This solution is  the union of two intervals  (,0) U (,).




Solved.





It is good to have the plot of the quadratic function before your eyes to check yourself at each step.











Plot y = 







If you want to see more solved problems/samples of this type, look into the lesson



    - Solving inequalities for rational functions with non-zero right side 



in this site.





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



    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.





The referred lesson is the part of this online textbook under the topic  "Inequalities".







---------

"josgarithmetic", having zero knowledge on the subject (and knowing about it), 

for what reason do you produce incorrect solutions in this forum?





There is a simple rule: work in that areas where you have enough knowledge,
and do not invade in the areas where you are not an expert.


And you will be happy . . . As well as all the others in this forum.




 

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