Question 63264
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solve the rational inequality (x-2)/(x+1) >= 3.
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<pre>
They want you solve this inequality

    {{{(x-2)/(x+1)}}} >= 3.    (1)


Transform it equivalently this way

    {{{(x-2)/(x+1)}}} - 3 >= 0             <<<---===  moving 3 from right side to left side with changing the sign

    {{{(x-2)/(x+1)}}} - {{{(3(x+1))/(x+1)}}} >= 0    <<<---=== writing '3' with the common denominator

    {{{((x-2) - 3(x+1))/(x+1)}}} >= 0       <<<---===  simplifying

    {{{(-2x -5)/(x+1)}}} >= 0               <<<---===  simplifying further



Now, the left side rational function can be non-negative if and only if

    EITHER the numerator is non-negative and denominator is positive

        -2x - 5 >= 0  and  x + 1 > 0    (2)

    OR     the numerator is non-positive and denominator is negative

        -2x - 5 <= 0  and  x + 1 < 0.   (3)



In case (2),  -2x >= 5  and  x > -1,  which is the same as  

              x <= -5/2 and  x > -1.

              These both inequalities, taken together, has no solution.



In case (3),  -2x <= 5  and  x < -1,  which is the same as  

              x >= -5/2 and  x < -1.


Thus the final solution to the given inequality is this set of real numbers  -5/2 <= x < -1,
or, in the interval notation, the set [{{{-5/2)}}},{{{-1}}}).
</pre>Solved.