your favorite online clothing store is offering a membership that allows you unlimited shipping throughout…

Explanation:

Step1: Set up an equation

Let $n$ be the number of orders. The cost without membership is a flat - rate of $$29.99$. The cost with membership is the base number of orders cost plus the per - order cost for additional orders. The base number of orders is $7.5$ (this seems to be a non - integer in the problem setup, but we'll work with it as given), and the per - order cost is $$3.99$. The cost with membership $C_m$ can be written as $C_m=3.99(n - 7.5)$ (assuming we are only considering the cost above the base number of orders). We want to find when $C_m<29.99$.

Step2: Expand and solve the inequality

First, expand the left - hand side of the inequality: $3.99n-3.99\times7.5 < 29.99$. Calculate $3.99\times7.5 = 29.925$. So the inequality becomes $3.99n-29.925 < 29.99$. Add $29.925$ to both sides: $3.99n<29.99 + 29.925$, so $3.99n<59.915$. Then divide both sides by $3.99$: $n<\frac{59.915}{3.99}\approx15.02$. Since $n$ represents the number of orders and we can't have a fraction of an order, and we want the membership to be worth it, we consider the smallest whole number of orders for which the membership is cheaper. If $n = 8$ (because we start counting from above the base number of $7.5$), the cost with membership for $n$ orders: Let's assume we start from the base number of orders. The cost with membership for $n$ orders is $3.99(n - 7.5)$ (for $n>7.5$). If $n = 8$, the cost is $3.99\times(8 - 7.5)=3.99\times0.5 = 1.995$. If we consider the general case, we want to find when the total cost with membership (including the base part) is less than the non - membership cost. Let $n$ be the total number of orders. The cost with membership $C = 3.99(n - 7.5)$ (for $n>7.5$). We want $3.99(n - 7.5)<29.99$. Solving for $n$: [ \begin{align*} 3.99n-29.925&<29.99\ 3.99n&<29.99 + 29.925\ 3.99n&<59.915\ n&<\frac{59.915}{3.99}\approx15.02 \end{align*} ] Since $n$ is the number of orders and must be a whole number, and we want the membership to be worth it, we consider the number of orders above the base number. The base number of orders is $7.5$. The number of additional orders $x$ such that the membership is worth it. Let's set up the equation $3.99x<29.99$ (ignoring the base number for simplicity, as we are interested in when the cost of additional orders is less than the non - membership cost). $x=\frac{29.99}{3.99}\approx7.52$. Since we are talking about whole orders, and we start from the base number of $7.5$ orders, the total number of orders $n$ (including the base) should be $8$ (because when $n = 8$, the cost with membership starts to be potentially less than the non - membership cost).

Answer:

8