Question 11·Medium·Linear Inequalities in One or Two Variables
A factory can manufacture a batch of widgets using one of two machines.
- Machine A requires a one-time setup fee of $1,200 and then costs $2.40 per widget to operate.
- Machine B has no setup fee and costs $3 per widget to operate.
Which inequality can be solved to find , the number of widgets for which the total cost of using Machine A is less than the total cost of using Machine B?
For inequality word problems like this, quickly turn each description into an algebraic expression: fixed fee plus (per-unit cost). Then carefully match the wording to an inequality symbol: "is less than" means use . Put the full expression for the first quantity on one side, the full expression for the second quantity on the other, and only then look for the answer choice that exactly matches that structure without rearranging or solving it.
Hints
Turn the words into algebraic expressions
First, write an expression for the total cost of each machine in terms of , the number of widgets. One of them has a fixed setup fee plus a per-widget cost; the other has only a per-widget cost.
Focus on the phrase "is less than"
The question says "the total cost of using Machine A is less than the total cost of using Machine B." Which inequality symbol ( or ) matches "is less than"?
Put the pieces together
Once you know each machine’s total cost expression and which symbol represents "is less than," place the Machine A expression, then the correct symbol, then the Machine B expression to form your inequality. Then look for the answer choice that matches that structure.
Desmos Guide
Enter the cost functions
In Desmos, type C_A(x) = 1200 + 2.40x for Machine A and C_B(x) = 3x for Machine B. These lines show the total cost of each machine for any number of widgets .
See which cost should be smaller
Look at the graph and think about the condition "Machine A is less than Machine B"—that means you want the -values of to be below the -values of . Translate that into an inequality of the form (expression for Machine A) (expression for Machine B), then choose the answer choice whose inequality matches that pattern.
Step-by-step Explanation
Write an expression for each machine’s total cost
Let be the number of widgets.
- Machine A: It has a setup fee of $1,200 and then costs $2.40 per widget, so the total cost is .
- Machine B: It has no setup fee and costs $3 per widget, so the total cost is .
Translate the verbal comparison into an inequality
The question asks for the number of widgets for which the total cost of using Machine A is less than the total cost of using Machine B.
In symbols, this means:
- (total cost using Machine A) (total cost using Machine B).
So you want an inequality where the expression for Machine A’s cost is on the left, the symbol is in the middle, and the expression for Machine B’s cost is on the right.
Form the inequality and match it to a choice
Substitute the cost expressions into the inequality structure from Step 2:
- Machine A’s cost:
- Machine B’s cost:
So the inequality is
.
Among the answer choices, this matches choice D.