 Hello and welcome to the session. In this session we will discuss a question which says that if x be real, show that value of 2x square minus 2x plus 4 whole upon x square minus 4x plus 3 cannot lie between minus 7 and 1. Now before starting the solution of this question, we should know about a method and that is the method of intervals. Now for this method, the first step is to factorize the quadratic expression whose coefficient of x square is positive and express 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. Secondly, plot the points alpha and beta to the number line and thus dividing the number line into three parts. After dividing the number line into three parts and plotting the points alpha beta starting from the very right region put the sign plus minus and plus so that the expression x minus alpha the whole into x minus beta the whole is non-negative in the region on the right of beta. Now when x minus alpha the whole into x minus beta the whole is greater than equal to 0 then the required range will be minus infinity is less than x is less than equal to alpha or beta is less than equal to x is less than infinity. It means that x minus alpha the whole into x minus beta the whole is positive in this region. Now this method will work out as a key idea for solving out this question and now we will start with the solution. Now here we have to show that this expression cannot lie between minus 7 and 1. Now let us take y is equal to 2x square minus 2x plus 4 whole upon x square minus 4x plus 3. Now cross multiplying this will be y into x square minus 4x plus 3 the whole is equal to 2x square minus 2x plus 4. Further this implies yx square minus 4xy plus 3y y is equal to 2x square minus 2x plus 4. Now taking all these terms on this side, this implies y minus 2 the whole into x square minus 2 into 2y minus 1 the whole into x plus 3y square minus 4 is equal to 0. Now this is a quadratic equation in x and since x is real, therefore, b square minus 4ac is greater than equal to 0. Now comparing this equation with the standard form of quadratic equation here, a is equal to y minus 2, b is equal to minus 2 into 2y minus 1 the whole and c is equal to 3y minus 4. Now b square minus 4ac is greater than equal to 0, so substituting the values of a, b and c here, we get minus 2 into 2y minus 1 the whole whole square minus 4 into y minus 2 the whole into 3y minus 4 the whole is greater than equal to 0. Further this implies our opening the square it will be 4 within brackets 4y square plus 1 minus 4y minus 4 and multiplying these two expressions it will give 3y square minus 10y plus 8 is greater than equal to 0. Now dividing both sides by 4 this implies 4y square plus 1 minus 4y minus within brackets 3y square minus 10y plus 8 is greater than equal to 0. This implies 4y square plus 1 minus 4y minus 3y square plus 10y minus 8 is greater than equal to 0. On solving this this implies y square plus 6y minus 7 is greater than equal to 0. Now factorizing this by splitting the middle term this will be y square plus 7y minus y minus 7 is greater than equal to 0 which will be further y within brackets y plus 7 and from these two terms minus 1 is common and within brackets y plus 7 is greater than equal to 0. This implies y plus 7 the whole into y minus 1 the whole is greater than equal to 0. Now putting each factor equal to 0 we get y is equal to minus 7 and 1. Now by the method of intervals we will plot these points on the number line. Now we have plotted these points on the number line. Now starting from the right put the signs as plus minus and plus as y plus 7 the whole into y minus 1 the whole is greater than equal to 0. So according to the formula which is given in the key idea if x minus alpha the whole into x minus beta the whole is greater than equal to 0 then the range is minus infinity is less than x is less than equal to alpha or beta is less than equal to x less than infinity. So here the range will be minus infinity is less than y is less than equal to minus 7 or 1 is less than equal to y is less than infinity as y plus 7 the whole into y minus 1 the whole is positive so the value of y will lie in this range as it is a positive region also as this region is negative so y cannot lie in between this region and this region varies from minus 7 to 1. So we can write that y cannot lie between minus 7 and 1 as y plus 7 the whole into y minus 1 the whole is positive or in other words you can say that the expression 2x square minus 2x plus 4 whole upon x square minus 4x plus 3 cannot lie between minus 7 and 1. So this is the solution of the given question and that's all for the session hope you all have enjoyed the session.