SOLUTION: Can someone help me with this? Solve the inequality algebraically: x^3-x^2-11x+3/2x^3-7x^2-19x+60 ≥ 0 Thank you.

Plot y =  (red) and y =  (green)


1.  The plots clearly say that the numerator has the root -3 and two other real roots that are not integer numbers.
    You divide the numerator by (x+3) (long division) and find the quotient polynomial which is quadratic.
    Then find the two roots of this quadratic. In this way you will find all the roots of the numerator.


2.  The plots say that the denominator has the roots -3, 4  and one other real root which is not integer number.
    You divide the denominator by (x+3)*(x-4) (long division) and find the quotient polynomial which is linear binomial.
    This linear binomial will give you the third root of the denominator. In this way you will find all the roots of the denominator.


3.  In the given rational function, write the numerator and denominator as the product of linear factors.


4.  The factor (x+3) will be common for the numerator and denominator.
    You can cancel this factor. It doesn't make influence on the solution set you are working on.
    As "anamnesis", this canceling will remain the hole x = -3 in the solution set.


5.  After canceling, the numerator and denominator will have two factors each.


6.  What to do next ? How to analyse this new rational function ?

    Read about it in the lesson

    - Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials 

    in this site.