Jessica and martha each have a bag of cookies with unequal quantities. they have 30 cookies total between the two of them. each of them ate 6 cookies from their bag. the product of the number of cookies left in each bag is not more than 80. how many more cookies will jessica have martha? if x represents the number of cookies jessica started with, complete the statements below. the inequality that describes the relationship between the number of cookies each one of them has is x2 - x + 224 ≥ 0. jessica has at least cookies more than martha.

If x represents the number of cookies jessica started with, then martha started with 30-x, because the sum of the cookies should be 30.  each of them ate 6 cookies, so jessica now has x-6 cookies, while martha now has 30-x-6 or 24-x cookies.  the product  of the number of cookies left in each bag is not more than 80. this is represented mathematically as (x-6)(24-x)  ≤ 80.so, we start from the previous inequality.      simplifying,  we have      subtract 80 from both sides      or, we can multiply -1 to all the terms to make the leading coefficient positive. this will change the sign of the inequality. that is        the inequality that describes the relationship between the number of cookies each one of them has is x²  - 30x + 224 ≥ 0.we can factor the inequality as (x-16)(x-14)≥0the inequality is satisfied when the two factors are both positive, or both negative.  the factor x-16 is negative if x is less than 16, and positive if it is greater than 16.the factor x-14 is negative when x is less than 14, and positive when x is greater than 14.  both of the factors are negative when x is less than or equal to 14, and both of them are positive when x is greater than or equal to 16.  since we are looking for a case where jessica, x, is more than martha, 30-x, we only need the solution where x is greater than or equal to 16.  therefore,  jessica should have at least 2 cookies more than martha.