Final answer:
To find the smallest number of signs that Mark can sell to have an income of $828 in one week, we need to solve a quadratic equation. The solution is x = 23, so Mark can sell 23 signs to achieve his desired income.
Step-by-step explanation:
To find the smallest number of signs that Mark can sell to have an income of $828 in one week, we need to set up an equation using the given information.
According to the problem, Mark sells 59 - x signs per week when he charges x dollars for each sign.
The income Mark generates in one week is equal to the number of signs sold multiplied by the price per sign. So the equation for Mark's income is:
Income = x * (59 - x)
To find the smallest number of signs that Mark can sell to have an income of $828, we need to solve the equation:
x * (59 - x) = 828
By expanding and rearranging the equation, we get:
x^2 - 59x + 828 = 0
This is a quadratic equation. We can solve it by factoring or by using the quadratic formula. Factoring the equation, we get:
(x - 23)(x - 36) = 0
Setting each factor equal to zero, we find two possible values for x: x = 23 and x = 36.
Since we are looking for the smallest number of signs, Mark can sell, we take the smaller value of x, which is x = 23.
Therefore, the smallest number of signs that Mark can sell to have an income of $828 in one week is 23 signs.