Write the total cost as a fixed charge plus the cost per item multiplied by the number of items. Set it less than or equal to the budget. If your answer is a decimal bound, choose the greatest whole number that satisfies the inequality.
The 2027 SEC G3 Mathematics K310 syllabus covers linear inequalities in one variable. This guide shows how to turn an algebraic bound into a feasible number of items and check that it is the largest possible.
- Define the variable and its allowed values.
- Use ≤ for a total that may equal the budget.
- Isolate the variable before choosing a whole number.
- Test the chosen number and the next one up.
CHAPTER 1 OF 4 · Back to contents
Separate the fixed charge from the item cost
A delivery fee is paid once. An item cost is paid for each unit. If delivery costs 18 dollars and each item costs 7 dollars, buying n items costs 18 + 7n dollars.
For a budget of 100 dollars, the cost may reach 100 but cannot exceed it. Write 18 + 7n ≤ 100. State that n is a non-negative integer in the model.
Solve the inequality before applying the whole-number restriction. This makes it clear whether the decimal or fraction is a mathematical bound or a possible purchase.
CHAPTER 2 OF 4 · Back to contents
Worked example | How many kits can the class buy?
Subtract 18 from both sides to obtain 7n ≤ 82. Divide both sides by positive 7 to obtain n ≤ 82/7, which is about 11.714. The largest permitted whole number is 11.
| Kits | Total cost | Within $100? |
|---|---|---|
| 11 | $95 | Yes |
| 12 | $102 | No |
Twelve is closer to 11.714 than eleven is, but its total is above the budget. The question asks for the largest feasible count, so ordinary rounding gives the wrong choice here.
The total increases by 7 dollars each time another kit is added. Once 12 fails, all greater counts fail as well.
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CHAPTER 3 OF 4 · Back to contents
Practise with a strict limit
A class pays a fixed 16 dollars plus 6 dollars per badge. It must spend less than 100 dollars. Then 16 + 6n < 100, so n < 14. The largest permitted whole number is 13.
At 14 badges the cost is exactly 100 dollars. If the wording changes to ‘at most 100 dollars’, 14 is allowed. Check whether the boundary itself is included.
On a number line, the strict inequality has an open endpoint at 14. In the badge context, only the permitted integer points below that endpoint are usable.
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CHAPTER 4 OF 4 · Back to contents
Frequently asked questions
Why does the inequality direction stay the same?
Subtracting the same amount and dividing by a positive number preserve the direction.
Is the largest count always the integer below the bound?
If an integer bound is included, use the bound itself. For a strict upper bound at an integer, use the next smaller integer.
What should I write on the final line?
Give the count with its unit or item name, such as 11 kits.
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