What you’ll learn
Model constraints using inequalities.
Key idea
Let a variable represent the unknown, express the fixed amount and per-unit amount, and use an inequality symbol that matches the wording. A budget of at most $20 means the total is less than or equal to 20.
After solving, consider what values make sense. A number of rides cannot be negative and is usually a whole number, even if the algebraic boundary is an integer. State the allowed range and answer the original question in context.
Worked example
Enter the exact value or ordered values in the format requested below.
Airtime x is a nonnegative real number of minutes, costing $2 per minute plus one $1 fee. The total must be at least $19. Create and isolate the inequality, retaining equality and domain.
- Cost is 2x+1, with airtime x>=0; 'at least' includes equality, giving 2x+1>=19.
- Subtract 1, then divide by positive 2 without changing direction: x>=9, within x>=0.
Answer: 2x+1>=19
Quick Check
Choose which candidate representation satisfies the requirements of 'Model constraints using inequalities' and justify the choice.
For nonnegative real airtime x at $2/minute plus one $2 fee, cost must be at most $18. Compare A: 2x+2<=18; B: 2x+2>=18; C: 2x+2<18. Select and justify the full allowed set.