For each quantity condition (total pieces, chairs, tables), decide whether "at least" means using $\ge$ or $\le$ in the inequality.

Go line by line in each answer choice and ask: Does this line correctly match one specific sentence from the problem? One sign error is enough to eliminate a choice.

Pick a sample combination that clearly meets the word conditions, such as $x=30$ chairs and $y=10$ tables (this gives 40 pieces and a total cost of $18(30) + 45(10)$). In Desmos, type 18*30 + 45*10 to confirm that this total is less than or equal to 1200.

For each answer choice, enter its inequalities into Desmos (e.g., for choice A: 18x + 45y >= 1200, x + y >= 40, x >= 20, y >= 5). Then add the sample point as (30,10) and see whether that point lies in the shaded region for that system. Compare across choices to see which system treats the spending limit and “at least” conditions in a way that includes this valid point.

In Desmos, look at the boundary line for the cost, 18x + 45y = 1200, and see which side is shaded for each system: the region below the line matches "no more than $1,200," while the region above would match "at least $1,200." Use this to decide which inequality direction correctly models the spending constraint.

We are told that $x$ is the number of chairs and $y$ is the number of tables.

• Each chair costs $18, so chairs cost $18x$ dollars.
• Each table costs $45, so tables cost $45y$ dollars.

So the total cost is $18x + 45y$.

The center can spend no more than $1,200.

• "No more than" means the total cost is less than or equal to $1,200.

• In symbols, that is the inequality

Any choice using $\ge 1200$ for this part would mean they must spend at least $1,200, which does not match the problem.

Next, use the phrases about how many items they need:

1. At least 40 pieces of furniture altogether:

   • Total pieces is $x + y$.
   • "At least 40" means $x + y \ge 40$.

2. At least 20 chairs:

   • Chairs are $x$, so "at least 20 chairs" means $x \ge 20$.

3. At least 5 tables:

   • Tables are $y$, so "at least 5 tables" means $y \ge 5$.

Any choice that uses $\le$ instead of $\ge$ for these minimum amounts is not matching "at least."

Putting all the constraints together, the system of inequalities that matches the word problem is:

• $18x + 45y \le 1200$ (no more than $1,200)
• $x + y \ge 40$ (at least 40 pieces of furniture)
• $x \ge 20$ (at least 20 chairs)
• $y \ge 5$ (at least 5 tables)

Comparing with the options, this system corresponds to choice D.