Question 1209291
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Solve the inequality 4t^2 \le -9t + 12.  Write your answer in interval notation.
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        The solution by the other tutor in his post is incorrect.

        The mistake is in incorrect factoring of the equation (which, actually, can not be factored).


        I came to bring a correct solution.



<pre>
Write in the standard form quadratic inequality

    4t^2 + 9t - 12 <= 0.    (1)


The corresponding quadratic equation is

    4t^2 + 9t - 12 = 0.


Find its roots using the quadratic formula

    {{{t[1,2]}}} = {{{(-9 +- sqrt(9^2 -4*4*(-12)))/(2*4)}}} = {{{(-9 +- sqrt(273))/8}}}.


The roots are  {{{(-9 - sqrt(273))/8}}}  and  {{{(-9 + sqrt(273))/8}}}.


The quadratic function in the left side of (1) is non-negative between the roots

    {{{(-9 - sqrt(273))/8}}} <= t <= {{{(-9 + sqrt(273))/8}}}


In the interval form, the solution set is the union of two sets

    ({{{-infinity}}},{{{(-9 - sqrt(273))/8}}}] U [{{{(-9 + sqrt(273))/8}}},{{{infinity}}}).    <U>ANSWER</U>
</pre>

Solved.