You want the first shop to be the cheaper option. How do you write "cost at first shop is less than cost at second shop" using your expressions?

After solving the inequality, you will get a decimal boundary for $n$. Ask yourself: must $n$ be a whole number, and if so, which whole number just passes that boundary?

In Desmos, type y = 40 + 2.35x for the first shop and y = 3.10x for the competing shop. These show total cost versus number of parts $x$.

Use Desmos to tap or click where the two lines intersect, or use the point-of-intersection feature. Note the $x$-coordinate of this intersection; this is the exact number of parts where the two costs are equal.

Because the first shop must be less expensive, you need a number of parts greater than the $x$-value at the intersection. Think about the smallest whole number of parts that is greater than this intersection $x$-value.

Let $n$ be the number of parts produced.

• Cost at the first shop: it charges a $40 set-up fee plus $2.35 per part, so the total cost is

• Cost at the competing shop: it charges $0 set-up fee and $3.10 per part, so the total cost is

We want the cost at the first shop to be less than the cost at the competing shop.

So write the inequality:

Isolate $n$ step by step.

Subtract $2.35n$ from both sides:

Now divide both sides by $0.75$ (a positive number, so the inequality sign stays the same):

Compute $40 \div 0.75$ to get a decimal value for the boundary.

Evaluating $40 \div 0.75$ gives approximately $53.33$, so the inequality is

This means $n$ must be greater than $53.33\dots$, not equal to it.

Since $n$ represents a number of parts, it has to be a whole number. The smallest whole number greater than $53.33\dots$ is 54, so the minimum number of parts is 54.