"Profit must be at least $20\%$ and at most $35\%$ of revenue" means profit lies between two multiples of revenue. How can you write that as one compound inequality using profit and revenue?

After you write $0.20(\text{revenue}) \le \text{profit} \le 0.35(\text{revenue})$, replace revenue and profit with your expressions in $g$ and $k$, then solve each side of the inequality separately to get a lower and an upper bound for $g$.

In Desmos, let $x$ stand for $g$. Enter R(x) = 45x for revenue and P(x) = 27x - k for profit. If you like, choose a convenient positive value for k (for example, type k = 90) so Desmos can graph the lines.

Enter L(x) = 0.2*R(x) to represent $20\%$ of revenue. Look at the intersection of the graphs of P(x) and L(x) and click it; the $x$-coordinate of this point is the minimum $g$ value that meets the $20\%$ requirement for your chosen $k$.

Enter U(x) = 0.35*R(x) to represent $35\%$ of revenue. Find the intersection of P(x) and U(x); the $x$-coordinate is the maximum $g$ value that keeps profit at or below $35\%$ of revenue for that same $k$.

Use Desmos to compute each choice’s bounds for the same $k$ (for example, type k/18, 4k/45, etc.) and compare these numbers with the two intersection $x$-values you found. The correct answer is the one whose lower and upper bounds match those intersection values.

• Revenue per gadget is $45$, so weekly revenue is $45g$.

• Variable cost per gadget is $18$, and fixed weekly cost is $k$, so total weekly cost is $18g + k$.

• Profit = revenue $-$ total cost, so

  $\text{profit} = 45g - (18g + k) = 27g - k$.

The company wants profit to be at least $20\%$ and at most $35\%$ of revenue.

• $20\%$ of revenue is $0.20 \cdot (45g)$.
• $35\%$ of revenue is $0.35 \cdot (45g)$.

"At least" becomes $\ge$ and "at most" becomes $\le$. So the condition is:

$0.20(45g) \le 27g - k \le 0.35(45g)$.

Start with the left side:

Compute $0.20 \cdot 45 = 9$:

Subtract $9g$ from both sides:

Add $k$ to both sides:

Divide by $18$ (positive, so the inequality direction stays the same):

So $g$ must be at least $k/18$.

Now use the right side:

Compute $0.35 \cdot 45 = 15.75$:

Subtract $15.75g$ from both sides:

Add $k$ to both sides:

Divide by $11.25$ (positive):

Since $11.25 = 45/4$, dividing by $11.25$ is the same as multiplying by $4/45$:

So $g$ must be at most $4k/45$.

Both conditions must be true at the same time:

• From the left inequality: $g \ge k/18$
• From the right inequality: $g \le 4k/45$

Together, this is the compound inequality:

This matches answer choice D.