Write an inequality for the total time using 12 minutes per bracelet and 18 minutes per necklace, and remember the phrase "at most 960 minutes" when choosing the inequality symbol.

Use the equation for total pieces to express one variable in terms of the other, and substitute into the time inequality so that you get an inequality involving only the number of necklaces.

Once you have an inequality in terms of the number of necklaces, solve it and think about what value you should choose to get the maximum number that still satisfies the inequality.

In Desmos, let the number of necklaces be $x$ and type the expression for total time: f(x) = 12*(60 - x) + 18*x. This gives the total minutes used when making $x$ necklaces and $(60 - x)$ bracelets.

Add the horizontal line y = 960 to represent the time limit. Look at where the graph of $f(x)$ meets the line y = 960; the $x$-value at this intersection is the largest number of necklaces that exactly uses all the available time.

Use a table in Desmos (click the gear next to f(x) and add a table) to plug in integer $x$-values near the intersection point. Find the greatest integer $x$ for which $f(x) \le 960$; that $x$ is the maximum number of necklaces.

Let:

• $b$ = number of bracelets
• $n$ = number of necklaces

We are told there are exactly 60 pieces in total, so:

Each bracelet takes 12 minutes and each necklace takes 18 minutes, and the jeweler has at most 960 minutes, so the time inequality is:

We want to maximize $n$ (the number of necklaces).

From the total pieces equation:

solve for $b$:

Now substitute $b = 60 - n$ into the time inequality $12b + 18n \le 960$:

Distribute the 12 in $12(60 - n)$:

Combine like terms $-12n + 18n$:

Now isolate $n$ by subtracting 720 from both sides:

Then divide both sides by 6:

Compute the division:

So the inequality becomes:

This means the number of necklaces cannot be more than 40. To maximize the number of necklaces, we choose $n = 40$.

Check quickly: if $n = 40$, then $b = 60 - 40 = 20$.

• Time for 20 bracelets: $20 \cdot 12 = 240$ minutes
• Time for 40 necklaces: $40 \cdot 18 = 720$ minutes
• Total time: $240 + 720 = 960$ minutes, which exactly meets the limit.

Therefore, the maximum number of necklaces that can be made is 40.