Question

Will choose Brainliest.
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 than 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 x^2- _ x + 224 ≥ 0
Jessica has at least
_ cookies more than Martha.

Answer 1

• That part is total cookies as per algebraic formula x^2-(a+b)x+ab=0

Hence

The inequality is

$\\ \rm\rightarrowtail x^2-30x+224>=0$

$\\ \rm\rightarrowtail (x-14)(x-16)>=0$

$\\ \rm\rightarrowtail x>=14,16$

Solution range

• [14,16]

Atleast means difference=16-14=2

Answer 2

Answer:

x² - 30x + 224 ≥ 0

Jessica has at least 2 cookies more than Martha

Step-by-step explanation:

Let x = number of cookies Jessica started with

Let y = number of cookies Martha started with

Given:

• They have 30 cookies in total between them.

⇒ x + y = 30

Given:

• 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.

⇒ (x - 6)(y - 6) ≤ 80

Rewrite x + y = 30 to make y the subject:

⇒ y = 30 - x

Substitute this into (x - 6)(y - 6) ≤ 80 and rearrange:

⇒ (x - 6)(30 - x - 6) ≤ 80

⇒ (x - 6)(24 - x) ≤ 80

Expand and simplify:

⇒ 24x - x² - 144 + 6x ≤ 80

⇒ - x² + 30x - 144 ≤ 80

⇒ - x² + 30x - 224 ≤ 0

Divide both sides by -1 (remembering to change the direction of the sign):

⇒ x² - 30x + 224 ≥ 0

Comparing the inequality we have calculated with the inequality given, the missing number is 30.

Solve the inequality by factoring:

⇒ x² - 30x + 224 ≥ 0

⇒ x² - 16x - 14x + 224 ≥ 0

⇒ (x² - 16x) + (-14x + 224) ≥ 0

⇒ x(x - 16) - 14(x - 16) ≥ 0

⇒ (x - 16)(x - 14) ≥ 0

Checking inequalities:

If x > 16 then (x - 16)(x - 14) > 0

If x < 16 then (x - 16)(x - 14) < 0

If x > 14 then (x - 16)(x - 14) < 0

If x < 14 then (x - 16)(x - 14) > 0

Therefore,

x ≥ 16  or  x ≤ 14

Given:

• x + y + 30

⇒ If x ≥ 16 then y ≤ 14

⇒ If x ≤ 14 then y ≥ 16

As Jessica (x) has MORE cookies than Martha (y), then only
If x ≥ 16 then y ≤ 14 is valid.

16 - 14 = 2

Therefore, Jessica has at least 2 cookies more than Martha