 Hello and welcome to the session. In this session we will discuss a question which says that determine the range of values of x for which x squared minus 3x plus 24 whole upon x squared minus 3x plus 3 is less than 4 x is real. Now before starting the solution of this question we should know about a method and that is the method of intervals. In this method the first step is to factorize the quadratic expression whose coefficient of x squared is positive express the left hand side of the inequality in the form x minus alpha the whole into x minus beta the whole where alpha is less than beta. In the second step plot the points alpha and beta on the number line and thus dividing the number line into three parts. Now after plotting the points alpha beta on the number line starting from the very right region put the signs of plus minus and plus. Now when x minus alpha the whole into x minus beta the whole is written as 0 then the required range is minus infinity is less than x is less than alpha or beta is less than x is less than infinity. So this method will work out as a key idea for solving out this question and now we will start with the solution. Here it is given that x squared minus 3x plus 24 whole upon x squared minus 3x plus 3 is less than 4. Here for completing the square of this expression we will use the method of completing the squares. So by the method of completing the squares we have to add and subtract the square of half the coefficient of x and by this we are getting x square minus 3x plus 3 is equal to x minus 3 by 2 whole square plus 3 by 4. So this can be written as x square minus 3x plus 24 whole upon x minus 3 by 2 whole square plus 3 by 4 is less than 4. Since x minus 3 by 2 whole square is positive being a square for all real values of x therefore the denominator x minus 3 by 2 whole square plus 3 by 4 is also positive thus here we can cross multiply making no effect on the inequality sign. Now cross multiplying we get x square minus 3x plus 24 is less than 4 into x square minus 3x plus 3 the whole. This implies x square minus 3x plus 24 is less than multiplying 4 inside it will be 4x square minus 12x plus 12 and on solving we get minus 3x square plus 9x plus 12 is less than 0. Now dividing both sides by minus 3 this implies x square minus 3x minus 4 is greater than 0 as we know that on dividing by a negative quantity the inequality reverses. Now splitting the middle term this implies x square minus 4x plus x minus 4 is greater than 0. This implies x within brackets x minus 4 plus 1 within brackets x minus 4 is greater than 0 which implies x plus 1 the whole into x minus 4 the whole is greater than 0. Now putting each factor equal to 0 we get x is equal to minus 1 and x is equal to 4. Now by the method of intervals we will plot these points on the number line. So we have plotted the points minus 1 and 4 on the number line. Now starting from the well right region put the signs of plus minus and plus. Now as x plus 1 the whole into x minus 4 the whole is greater than 0 then we can find the range by using the formula given in the t-idea which is when x minus alpha the whole into x minus beta the whole is greater than 0 then the range is minus infinity is less than x is less than alpha or beta is less than x is less than infinity. So here alpha is minus 1 and beta is 4 so the required range is minus infinity is less than x is less than minus 1 or 4 is less than x is less than infinity. So this is the solution of this question and that's all for this session. Hope you all have enjoyed this session.