A reliable method
Isolate the variable, reverse the inequality when dividing by a negative number, and apply the context restriction last.
The common trap
Do not round a maximum affordable count up. Use the greatest integer that still satisfies the original constraint.
Make the connection
Translate “at most” as ≤ and “at least” as ≥. Test a boundary value and a value just outside it.
Separate a boundary from a feasible count
With a $7 fixed fee and $4 per item, a $30 budget gives 7+4n≤30, hence n≤23/4=5.75. Whole items restrict n to integers, making 5 the maximum. Five items cost $27 and six cost $31, so testing the neighboring integer verifies the maximum rather than merely finding a feasible value.
Track a negative multiplier
For −3x+2<11, subtract 2 and divide by −3: x>−3. The boundary −3 is excluded; the least integer is −2. Test it: 8<11 is true. The value −4 gives 14<11, false. These tests catch failure to reverse the sign without memorizing a rule in isolation.
Try, then check
Find the smallest integer satisfying x≥2 and 2x>7. The second condition gives x>3.5, so the answer is 4. Choosing 2 uses only the first condition. Write the intersection before rounding; otherwise two individually reasonable operations may select a value that violates one constraint.
What is the greatest integer x satisfying 3x + 4 ≤ 20?
- Subtract 4: 3x ≤ 16.
- Divide by positive 3: x ≤ 16/3.
- The greatest permitted integer is 5.
Compare the other answer choices
Discarded the largest feasible integer unnecessarily.
Rounded a noninteger upper boundary up, making it infeasible.
Reported the real boundary despite the integer requirement.
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.
Official mathematics skillsSiyavula Grade 10 mathematics