Let's analyze the problem step by step based on the provided information:
1. Let [tex]\( x \)[/tex] be the number of cookies Jessica started with.
2. Since the total number of cookies is 30, Martha started with [tex]\( 30 - x \)[/tex] cookies.
After eating 5 cookies each, the number of cookies left for Jessica is [tex]\( x - 5 \)[/tex], and for Martha is [tex]\( 30 - x - 5 = 25 - x \)[/tex].
The problem states that the product of the number of cookies left must be no more than 80:
[tex]\[
(x - 5) \times (25 - x) \leq 80
\][/tex]
Let's expand and simplify this inequality:
[tex]\[
(x - 5)(25 - x) \leq 80
\][/tex]
[tex]\[
25x - x^2 - 125 + 5x \leq 80
\][/tex]
[tex]\[
-x^2 + 30x - 125 \leq 80
\][/tex]
Subtract 80 from both sides to get a standard quadratic inequality:
[tex]\[
-x^2 + 30x - 205 \leq 0
\][/tex]
Multiply by -1 to make the inequality more familiar:
[tex]\[
x^2 - 30x + 205 \geq 0
\][/tex]
Therefore, the inequality that describes the relationship between the number of cookies is:
[tex]\[
x^2 - 30x + 205 \geq 0
\][/tex]
After eating 5 cookies each, we need to determine how many more cookies Jessica has compared to Martha. This is the difference:
Given the results:
Discriminant: [tex]\( 80 \)[/tex]
Square root of the discriminant: [tex]\( 8.94427190999916 \)[/tex]
Root 1: [tex]\( 19.47213595499958 \)[/tex]
Root 2: [tex]\( 10.52786404500042 \)[/tex]
Take [tex]\( x = 19.47213595499958 \)[/tex]:
[tex]\[
y = 30 - x = 30 - 19.47213595499958 = 10.52786404500042
\][/tex]
Jessica's remaining cookies:
[tex]\[
x - 5 = 19.47213595499958 - 5 = 14.47213595499958
\][/tex]
Martha's remaining cookies:
[tex]\[
y - 5 = 10.52786404500042 - 5 = 5.52786404500042
\][/tex]
The excess cookies Jessica has compared to Martha after eating 5 cookies each is:
[tex]\[
14.47213595499958 - 5.52786404500042 = 8.94427190999916
\][/tex]
So, Jessica has at least [tex]\( 8.94427190999916 \)[/tex] cookies more than Martha.
Thus, the completed statements are:
The inequality that describes the relationship between the number of cookies each one of them has is [tex]\( x^2 - 30x + 205 \geq 0 \)[/tex].
Jessica has at least [tex]\( 8.94427190999916 \)[/tex] cookies more than Martha.
