BTT Mathematics / Primary Mathematics Learning Hub / Inequalities and Feasible Ranges
An equation identifies values that make two quantities equal. An inequality identifies values that lie on one side of a boundary or inside a range. The symbols <, >, ≤ and ≥ are not decorative versions of the equals sign. They describe ordering conditions that may admit many answers.
If x<7 and x is a positive whole number, the possible values are 1,2,3,4,5,6. If x≤7, the boundary value seven is also allowed. The difference between an open and closed boundary is small in notation and large in meaning.
This guide develops number-line representation, one-step inequalities, compound constraints, whole-number feasible ranges, upper/lower bounds and optimisation-style boundary reasoning. It is Primary Mathematics enrichment that connects Bounds, Extremes and Guaranteed Conclusions, Simultaneous Conditions and Logic, Deduction and Truth Conditions.
Formal symbolic manipulation beyond the stated examples is optional extension. Use the MOE Primary curriculum page and the learner’s actual school programme for required scope.
Meaning and boundaries · Number lines · Solve simple inequalities · Compound conditions · Whole-number ranges · Feasible regions · 24 questions · Worked answers · Teaching and transfer
1. Read the boundary before calculating
The statement x>5 says x is greater than five. It does not include five. The statement x≥5 says x is at least five, so five is included. Similarly, x<9 excludes nine while x≤9 includes it.
Worked example A: Positive whole numbers below seven
If x is a positive whole number and x<7, then x∈{1,2,3,4,5,6}. There are 6 possible values.
Worked example B: Whole numbers from zero
If x is a whole number and x≤4, then x can be 0,1,2,3,4. The domain matters: including zero changes the solution set from a positive-whole-number interpretation.
Worked example C: “Between” needs clarification
“Between three and eight” may be used informally in different ways. A well-formed mathematical statement should say 3<x<8, 3≤x≤8 or another explicit boundary combination. Do not infer inclusion from conversational wording when the symbols are available.
Worked example D: At least and at most
“At least twelve” means x≥12. “At most twelve” means x≤12. “More than twelve” means x>12. “Fewer than twelve” means x<12.
2. Number lines make inclusion visible
For x>3, mark an open point at three and shade to the right. For x≥3, use a closed point at three and shade to the right. The open point means the boundary is not a solution; the closed point means it is included.
Worked example E: 2≤x<6
Mark two closed, six open and shade between them. If x is a whole number, the allowed values are 2,3,4,5.
Worked example F: x<−1
Mark an open point at −1 and shade left. Values such as −2 and −5 satisfy the inequality; −1 does not.
Direction is part of the meaning
On a standard horizontal number line, larger numbers lie to the right. This provides a visual check: x>4 should not be shaded left.
3. Preserve the order when adding or subtracting the same amount
If x+4<11, subtract four from both sides to obtain x<7. Adding or subtracting the same number preserves the order relationship.
Worked example G: Addition
x+9≤15 gives x≤6.
Worked example H: Subtraction
x−5>8 gives x>13.
Worked example I: Positive multiplication
3x≤18 gives x≤6 when x is a non-negative quantity, because dividing both sides by the positive number three preserves the order.
Optional extension: multiplying by a negative reverses order
For later-study bridge work, −2x<8 becomes x>−4 when both sides are divided by −2. The direction reverses. A simple number check explains why: x=0 satisfies the original because 0<8 and also satisfies 0>−4.
This reversal is optional enrichment, not required for the remainder of the guide. Most practice questions use positive multipliers and additive constraints.
Substitution checks a candidate
To test whether x=5 satisfies 2x+1≤12, substitute: 11≤12 is true. To test x=6, 13≤12 is false. One successful substitution checks one candidate; it does not solve an unrestricted inequality by itself.
4. Compound inequalities keep two boundaries active
The statement 4≤x≤10 means x is at least four and at most ten. A value must satisfy both conditions simultaneously.
Worked example J: Whole-number solutions
If x is a whole number and 4≤x<9, the solutions are 4,5,6,7,8.
Worked example K: Two separate conditions
x>2 and x≤7 combine to 2<x≤7. For integer x, the values are 3,4,5,6,7.
Worked example L: Inconsistent conditions
x>8 and x<5 cannot both hold. The feasible set is empty. The correct conclusion is no solution, not a guessed midpoint.
Worked example M: Redundant conditions
x≥10 and x≥4 reduce to x≥10. The second condition adds no new restriction because every value satisfying x≥10 already satisfies x≥4.
5. Whole-number domains turn intervals into finite candidate sets
If the problem says x is an integer, a bounded interval can be counted. If x is a real number, infinitely many values may lie between two boundaries.
Worked example N: Count integers
For integer x with −2≤x≤4, the values are −2,−1,0,1,2,3,4. There are 7 integers.
Worked example O: Positive whole numbers
For positive whole number n with 6<n≤12, the values 7 through 12 give 6 possibilities.
Worked example P: Maximum and minimum feasible values
If integer k satisfies 15≤k<22, the minimum is 15 and maximum is 21. The upper boundary 22 is excluded.
6. A feasible range is the overlap of every active constraint
A packing rule says each box may hold at most twelve items. A delivery requires at least nine items per box. The feasible whole-number counts per box are 9,10,11,12.
Worked example Q: Budget and minimum quality
A purchase quantity q must be at least five and at most eleven. The feasible set is 5≤q≤11. If q must also be even, the candidates are 6,8,10.
Worked example R: Capacity after a fixed amount
A container holds at most fifty litres and already contains eighteen. If x litres are added, 18+x≤50, so x≤32. If x is non-negative, the feasible range is 0≤x≤32.
Worked example S: Guaranteed comparison
If A≥30 and B≤24, then A is definitely at least six greater than B. The smallest possible difference occurs at A=30 and B=24. Boundary reasoning gives a guaranteed conclusion without knowing exact values.
Worked example T: A range can remain non-unique
If 20≤x≤30 and x is even, the candidates are 20,22,24,26,28,30. Additional information would be needed for a unique value. A range answer is not incomplete when the question’s data genuinely permit several possibilities.
7. Practice: 24 original questions
Use the stated domain in each question. “Whole number” includes zero; “positive whole number” begins at one.
Questions 1–8: Boundaries and number lines
1. List all positive whole numbers x satisfying x<7.
2. List all whole numbers x satisfying x≤4.
3. List all integers x satisfying 2≤x<6.
4. Translate “at least 12” into an inequality.
5. Translate “fewer than 9” into an inequality.
6. Does x=5 satisfy x>5?
7. Does x=5 satisfy x≥5?
8. For integers x with −2≤x≤4, how many values are possible?
Questions 9–16: Solve and combine conditions
9. Solve x+4<11.
10. Solve x+9≤15.
11. Solve x−5>8.
12. Solve 3x≤18.
13. Combine x>2 and x≤7. List the integer solutions.
14. Combine x>8 and x<5. What is the solution set?
15. If x≥10 and x≥4, which condition determines the final lower bound?
16. For positive whole number n with 6<n≤12, how many values are possible?
Questions 17–24: Feasible ranges and guarantees
17. Integer k satisfies 15≤k<22. Find the minimum and maximum possible values.
18. q is an even whole number with 5≤q≤11. List all possible q.
19. A container holds at most 50 L and already contains 18 L. If x≥0 litres are added, find the feasible range of x.
20. A class must contain at least 16 and at most 24 pupils. If the class size must be a multiple of 4, list the feasible sizes.
21. A≥30 and B≤24. What is the smallest guaranteed value of A−B?
22. x is an even whole number with 20≤x≤30. List all possible values.
23. A score s must satisfy 60≤s≤100. Does s=59 satisfy the condition? Does s=100?
24. A machine accepts integer input n only when 3<n<9. How many accepted integer inputs are there?
8. Worked answers
Answers 1–8
1. 1,2,3,4,5,6. Seven is excluded by the strict boundary.
2. 0,1,2,3,4. Whole numbers include zero and ≤ includes four.
3. 2,3,4,5. Two is included; six is excluded.
4. x≥12. “At least” includes the boundary.
5. x<9. “Fewer than” excludes nine.
6. No. Five is not greater than five.
7. Yes. Greater than or equal includes the boundary.
8. 7 values. −2,−1,0,1,2,3,4.
Answers 9–16
9. x<7. Subtract four from both sides.
10. x≤6. Subtract nine.
11. x>13. Add five.
12. x≤6. Divide by positive three.
13. 3,4,5,6,7. The combined interval is 2<x≤7.
14. No solution. No number is simultaneously greater than eight and less than five.
15. x≥10. It is the stronger lower bound.
16. 6 values. 7,8,9,10,11,12.
Answers 17–24
17. Minimum 15; maximum 21. Twenty-two is excluded.
18. 6,8,10. These are the even values in the interval.
19. 0≤x≤32. From 18+x≤50 and the non-negative condition.
20. 16,20,24. These multiples of four lie in the allowed range.
21. 6. The smallest difference is 30−24.
22. 20,22,24,26,28,30.
23. 59: no. 100: yes. The lower boundary excludes 59; the upper boundary includes 100.
24. 5 inputs. The integers are 4,5,6,7,8.
9. Teaching and transfer
If a learner treats an inequality like an equation and searches for one answer, ask for two different values that both satisfy x<7. The idea of a solution set becomes visible immediately.
When boundaries are confused
Use a number line with removable open and closed endpoint markers. Ask whether the boundary itself should be accepted before shading the rest of the range.
When combined conditions are handled separately
Draw each condition on the same number line and keep only the overlapping shaded region. This turns “and” into a visible intersection of allowed values.
When a range is mistaken for uncertainty or failure
Use a problem whose information genuinely permits several whole numbers. A complete answer can be a list, interval or count of possibilities. Mathematical precision does not require inventing a unique value.
When a guarantee is needed
Push quantities to the boundary that makes the claimed conclusion hardest to satisfy. This connects inequality reasoning to worst-case arguments in Bounds, Extremes and Guaranteed Conclusions.
Continue through this enrichment collection
For rule-based input and output restrictions, use Function Machines, Input–Output Tables and Inverse Rules. For explicit procedures and branches, use Mathematical Algorithms, Flowcharts and Decision Trees. For spatial regions defined by distance conditions, use Loci, Distance Constraints and Grid Geometry.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Domains and boundary conventions are stated locally; no official examination requirement is implied.