A laboratory must purchase containers of two chemical solutions, X and Y, for an experiment. Each container of solution X costs $40 and contains 3 grams of a required enzyme. Each container of solution Y costs $60 and contains 5 grams of the enzyme. The laboratory's purchasing budget for these solutions is at most $1,200, and the laboratory needs a minimum of 100 grams of the enzyme.
If is the number of containers of solution X purchased and is the number of containers of solution Y purchased, which of the following systems of inequalities represents the constraints on and ?
A system of inequalities describes several limits that must all hold at once. For a purchasing problem, multiply each amount per container by its container count, then add. A budget caps spending, while a minimum requirement sets a floor, so the inequalities point in opposite directions. Container counts also cannot be negative.
Hints
- Hint 1
A budget limits total spending. Multiply each solution's price by its number of containers, then add the costs. If the lab may spend less than its budget, which direction should the inequality point?
- Hint 2
An enzyme requirement limits the total grams from below. Multiply the grams in each container by the number purchased. A minimum allows exactly the required amount, but does it also allow more?
- Hint 3
Nonnegative means zero or greater. The lab could buy zero containers of one solution, but it cannot buy a negative number. What inequalities say that for both counts?
Step-by-step
Translate each limit
Step 1Write the spending limit
No Desmos needed. You're translating word limits, not solving an equation. The containers of X cost dollars, and the containers of Y cost dollars. Add those costs. At most $1,200 means the lab may spend less than or exactly that amount, so:
- Step 2
Write the enzyme requirement
The X containers provide grams of enzyme, and the Y containers provide grams. The minimum of grams means at least grams, so:
A spending cap sets an upper limit; a required minimum sets a lower limit.
- Step 3
Restrict the container counts
A container count is nonnegative, meaning zero or greater. Add and to the two limits. The system that represents the lab's purchasing constraints is:
Choice D.