 Hello and welcome to the session. In this session we will discuss the values that cause rational equations to be undefined. Let us recall the definition of rational expression. Rational expression is of the form p of x upon q of x where p of x and q of x are polynomials in variable x and q of x is not equal to 0. Now we will find the values of x for which a rational expression becomes undefined. A rational expression becomes undefined for those values of x where denominator is equal to 0. Let us consider the following rational expression. x minus 3 whole upon 2x plus 3 here numerator that is p of x is given as x minus 3 and denominator that is q of x is given as 2x plus 3. This expression becomes undefined when its denominator will be equal to 0. That is 2x plus 3 is equal to 0. Now we solve it for x and it implies that 2x is equal to minus 3 which further implies that x is equal to minus 3 by 2. So the value of x for which the rational expression x minus 3 whole upon 2x plus 3 becomes undefined is x is equal to minus 3 by 2. Now we find the values that cause a rational equation to be undefined. A rational equation is undefined at that value where the least common denominator is equal to 0. Now if in the solution we get a value where one of the expressions given in the equation becomes undefined or the least common denominator is equal to 0. Then we omit that value from the answer set and that value is called extremist solution. Let us consider this rational equation that is x upon x minus 3 minus 4 upon x minus 9 is equal to 3x minus 51 whole upon x square minus 12x plus 27. First of all we will simplify this quadratic by factorization. By factorization we can write this quadratic as x minus 3 the whole into x minus 9 the whole and we get this equation. Now here the least common denominator of all the denominators of this equation is x minus 3 the whole into x minus 9 the whole. So we multiply both sides of the equation by least common denominator that is x minus 3 the whole into x minus 9 the whole. We get x minus 3 the whole into x minus 9 the whole into x upon x minus 3 minus x minus 3 the whole into x minus 9 the whole into 4 upon x minus 9 is equal to x minus 3 the whole into x minus 9 the whole into 3x minus 51 whole upon x minus 3 the whole into x minus 9 the whole. Now this implies that now here x minus 3 the whole cancels with x minus 3 the whole. So here we are left with x into x minus 9 the whole minus. Now here x minus 9 the whole cancels with x minus 9 the whole and here we are left with 4 into x minus 3 the whole and this is equal to now here we are left with 3x minus 51. Now simply find this further we get x into x that is x square minus x into 9 is 9x so we have x square minus 9x. Now minus 4 into x is minus 4x minus 4 into minus 3 is plus 12 and this is equal to 3x minus 51 which implies that x square minus 13x plus 12 is equal to 3x minus 51 which further implies that x square minus 13x minus 3x plus 12 plus 51 is equal to 0 which implies that x square minus 16x plus 63 is equal to 0. Now we solve this quadratic for x by factorization and here we get x square minus 16x plus 63 can be written as x minus 7 the whole into x minus 9 the whole. So here we have x minus 7 the whole into x minus 9 the whole is equal to 0 which implies that x is equal to 7 and 9. So now we have two solutions of the given equation and these are x is equal to 9 and x is equal to 7. Now here we should note that x is equal to 9 makes the least common denominator equal to 0 that is if we put the value of x as 9 in least common denominator we get its value as 0. So we can say that the equation will become undefined at x is equal to 9 so it does an extraneous solution and we will omit it from the solution set. So x is equal to 7 is the only solution of the given rational equation. To solve our rational equation we follow the following steps. First we find the least common denominator of all the denominators. Then we multiply both sides of the equation by the least common denominator. Then we solve for x and lastly we take for extraneous solution that is those values of x where the equation becomes undefined that is those values of x where the least common denominator is equal to 0. This completes our session. Hope you enjoyed this session.