How many items must she sell to break even? Enter the correct answer.
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20
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Pilar is starting a small business that
makes specialized stationery. She finds
that producing larger quantities of a
design saves her money. Each print run
costs $50 plus $2 for each item. If y
dollars is the cost and x is the number of
items, this statement can be represented
by y = 2x + 50. She can sell each item for
$4.50. So her revenue, y, is equal to 4.5
times the number of items: y = 4.5x. The
solution of this system of equations is
her break-even point.

20

Step-by-step explanation:

We have two simultaneous equations.

(1)  y = 2x + 50

(2) y = 4.5x

1. Subtract (1) from (2)

0 = 2.5x - 50

2. Add 50 to each side

2.5x = 50

3. Divide each side by 2.5

x = 20

Pilar must sell 20 items to break even.

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.