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All courses Algebra I · A-CED.3 3 of 76
Represent constraints with equations, inequalities, and systems; interpret viable and non-viable solutions in context.

Write a single inequality constraint for a budget, capacity, or time limit

Problem
Shirts cost \(\$12\) each, and the spending limit is \(\$60\). Let \(s\) be the number of shirts. Write an inequality representing the budget.
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Hint

Multiply the cost per shirt by the number of shirts, then match “at most” with the correct inequality symbol.

“At most” means the total can be less than or equal to the limit.

Solution walkthrough

01

Define the count and resource

\[s~=~\text{number}~\text{of}~\text{shirts};~\text{budget}~\text{limit}~=~60~\text{dollars}\]

The prompt assigns s to the shirt count and gives 60 dollars as the maximum permitted spending.

02

Build the total-cost expression

\[\text{total}~\text{cost}~=~(\text{cost}~\text{per}~\text{shirt})(\text{number}~\text{of}~\text{shirts});~\text{total}~\text{cost}~=~12s~\text{dollars}\]

Each shirt costs 12 dollars, so multiplying that rate by s shirts gives the resource used.

03

Translate the limit language

\[\text{at}~\text{most}~60~\text{means}~\text{total}~\text{cost}~\le~60\]

At most includes amounts below the limit and the limit itself, so the comparison is less than or equal to rather than greater than or equal to.

04

Write the budget constraint

\[12s~\le~60\]

Substitute the cost expression for total cost. Both sides are dollar amounts, and the inequality prevents spending from exceeding 60 dollars.

05

Check the inclusive boundary and answer

\[s=5:~12(5)=60,~\text{so}~12s~\le~60~\text{is}~\text{true}\]

Five shirts exactly use the budget and must be allowed by at most. This confirms the inequality direction and choice A.

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Another way

  1. Divide the budget by the unit price to see s <= 5, then multiply both sides by positive 12 to recover the equivalent budget constraint 12s <= 60.

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Common mistake

Do not use >=. At most describes a ceiling: spending may equal 60 dollars but may not go above it.

Solution walkthrough video