Choose a method
Solve the boundary equation first, then retain the correct side. Reverse the inequality when multiplying or dividing by a negative number.
Avoid this trap
Strict boundaries exclude equality. A whole-number count must be rounded toward the feasible region, not automatically to the nearest integer.
Check and transfer
Test the last feasible integer and the next integer in the original condition. For two restrictions, use the overlap of their solution sets.
Worked example with a numerical check
A club has $43 for supplies. Each notebook costs $5. What is the greatest whole number of notebooks the club can buy? Divide the available amount by five to find the capacity boundary. A fraction of a notebook cannot be purchased, so take the integer below that boundary. The requested value is 8.
Contrast and diagnose
What is the least integer x satisfying −2x < −8? Dividing both sides by negative two reverses the inequality. The boundary is excluded, so take the next larger integer. The requested value is 5. An incorrect approach: Include the excluded boundary.
Transfer practice
A delivery must weigh at least 14 kg. Each identical box weighs 3 kg. What is the fewest whole boxes needed? Try first. Check: 5. Divide the minimum weight by the weight of one box. Round upward because the requirement is a minimum.
A club has $43 for supplies. Each notebook costs $5. What is the greatest whole number of notebooks the club can buy?
- Divide the available amount by five to find the capacity boundary.
- A fraction of a notebook cannot be purchased, so take the integer below that boundary.
- The requested value is 8.
Compare the other answer choices
Round upward although a budget is an upper limit.
Remove the remainder but forget the unit cost.
Treat the budget as notebook count.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications