 Hello and welcome to the session. In this session we will discuss a question which says that solve 2x square minus 11x is greater than or equal to 6 and we will recall it. Now let us start with the solution of the given question. Now here we want to solve this quadratic inequality that is 2x square minus 11x is greater than or equal to 6. Now to find its solution we follow the three steps. In step one we write its related equation that is 2x square minus 11x is equal to 6. Now in second step solve this quadratic equation. For critical value now we have the quadratic equation 2x square minus 11x is equal to 6. Now subtracting 6 from both sides of this equation we have 2x square minus 11x minus 6 is equal to 6 minus 6. This implies 2x square minus 11x minus 6 is equal to 0. Now let us factorize this quadratic equation by splitting the middle term. So this implies 2x square minus 12x plus x minus 6 is equal to 0. This further implies now from these two terms taking 2x common it will be 2x into x minus 6 is equal to 0. And from these two terms taking 1 common it will be plus 1 into x minus 6 is equal to 0. This implies x minus x the whole into 2x plus 1 the whole is equal to 0. Now either x minus 6 is equal to 0 or 2x plus 1 is equal to 0. So we have x is equal to switch or x is equal to minus 1 upon 2. Now we have got the critical values of x and let us start with third step. Now in the third step we will plot these critical values on the number line. Now we have got the critical values as 6 and minus 1 by 2. So here on this number line we have plotted the points minus 1 by 2 and 6. Now since in the given inequality we have the sign of greater than equal to. So here we have put dark circles to represent the points minus 1 by 2 and 6 which shows that we have also included the points minus 1 by 2 and 6. Now the points x is equal to minus 1 by 2 and x is equal to 6 divide the number line in three parts. In first part we have the interval x is less than minus 1 by 2 that is this yellow shaded portion. In the second part we have the interval minus 1 by 2 is less than x is less than 6 that is this green shaded portion. And in the third part we have the interval x is greater than 6 that is this pink shaded portion. Now we check for each interval which satisfies the given inequality. Now the given inequality is 2x square minus 11x is greater than equal to 6. Now subtract 6 from both sides so we have 2x square minus 11x minus 6 is greater than equal to 6 minus 6. Which implies 2x square minus 11x minus 6 is greater than equal to 0. First of all let us consider this interval that is the interval x is less than minus 1 by 2. And we can write this interval as open interval minus infinity to minus 1 by 2. Now let us take every point in this interval. So let us take x is equal to minus 1 that lies in this given interval. Now let us put this value in this inequality. So we have 2 into minus 1 whole square minus 11 into minus 1 minus 6 is greater than equal to 0. This implies now minus 1 whole square is 1 and 2 into 1 is 2 minus 11 into minus 1 is plus 11 minus 6 is greater than equal to 0. Now this implies now 11 plus 2 is 13 and 13 minus 6 is 7. So 7 is greater than equal to 0. This is true thus this interval the solution such of the given inequality. Now let us consider the second interval that is minus 1 by 2 is less than x is less than 6. Open interval minus 1 by 2 to 6. Now let us take every value lying in this interval. So let us take x is equal to 0 that lies in this interval. Now put x is equal to 0 in the given inequality. So we have 2 into 0 square minus 11 into 0 minus 6 is greater than equal to 0. Which implies now 2 into 0 will be 0 minus 11 into 0 is again 0 minus 6 is greater than equal to 0. Which implies minus 6 is greater than equal to 0 which is false as minus 6 is less than 0. Thus this interval is not the solution such of the given inequality. Now let us consider this interval that is x is greater than 6. Or we can write this interval as open interval 6 to infinity. Now let us take any value lying in this interval. So let us take x is equal to 7. Now put x is equal to 7 in the given inequality. So we have 2 into 7 square minus 11 into 7 minus 6 is greater than equal to 0. Now this implies now 7 square is 49 and 49 into 2 is 98 minus 11 into 7 is 77 minus 6 is greater than equal to 0. Which implies 98 minus 83 is greater than equal to 0 which further implies 15 is greater than equal to 0. Which is true thus this interval is the solution such of the given inequality. Thus we have solution set of the given inequality as x is less than minus 1 by 2 and x is greater than 6. Now we have the solution set as x is less than minus 1 by 2 plus greater than 6. So its solution will be union these two intervals now is 1 by 2 and x is equal to 6. Also satisfy the given inequality because we have greater than equal to sign. So this is the solution of the given inequality and here we have represented it on this number line. Thus the solution set of the given inequality is given interval minus infinity to minus 1 by 2 union 7 plus interval 6 to infinity. And this is the required answer of the given question. So this completes our session. Hope you all have enjoyed the session.