SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 14
An equation identifies equality. An inequality describes order. The symbols <, >, ≤ and ≥ tell us that one quantity is smaller, larger, no greater than or no less than another. They often describe not one answer but a whole set of possible values.
That change matters. Solving 2x + 3 = 11 gives one value, x = 4. Solving 2x + 3 < 11 gives every value less than 4. The arithmetic may look similar, but the meaning of the answer is different.
This guide develops inequality notation, number-line representation, algebraic transformations, sign reversal when multiplying or dividing by a negative number, compound constraints and contextual interpretation. Return to the Secondary Mathematics Hub. Revisit Equations and Equality for the balancing principle that inequalities extend.
Navigate: language · number lines · solving · negative reversal · compound inequalities · context · practice · answers.
1. Read the symbol as a sentence
x < 5 means x is less than 5. x > 5 means x is greater than 5. x ≤ 5 means x is at most 5. x ≥ 5 means x is at least 5.
The open side of the symbol faces the larger value. But do not rely only on a visual memory trick. Read the statement aloud.
Worked contrast
3 < 7 and 7 > 3 express the same order from opposite directions.
If the variable is on the right, such as 6 > x, it means the same as x < 6.
2. Strict and inclusive inequalities are different
x < 4 excludes 4. x ≤ 4 includes it.
On a number line, a strict endpoint is often shown with an open circle while an inclusive endpoint is shown with a filled circle, following the convention used in many school texts.
Context example
“Fewer than 30 students” means n < 30. “No more than 30 students” means n ≤ 30.
One word can determine whether the boundary value is allowed.
3. A number line displays a solution set
The solution x > 2 contains infinitely many real values: 2.1, 3, 100 and so on. A number line shows the boundary at 2 and shades or marks the allowed direction to the right.
Worked examples
x ≤ −1 uses a filled endpoint at −1 and extends left.
x < 3 uses an open endpoint at 3 and extends left.
Direction check
Choose a test value. For x > 2, try 5. Since 5 > 2 is true, the shaded side must include 5.
4. Inequalities can be compared through position
On a standard number line, values increase to the right. Therefore −2 > −5 even though 5 has greater magnitude than 2.
This connects inequalities to directed-number sense. A learner who misorders negative numbers will often struggle with inequality graphs before any algebra begins.
Diagnostic repair
If −7 < −3 feels uncertain, mark both on the number line. The left-right position decides order; distance from zero answers a different question.
5. Adding or subtracting the same quantity preserves order
If a < b, then a + c < b + c. Shifting both quantities by the same amount does not change which is larger.
Worked example
Solve x + 7 < 12. Subtract 7 from both sides: x < 5.
Check with x = 4: 4 + 7 = 11 < 12. Check the boundary x = 5: 12 < 12 is false, so 5 is correctly excluded.
6. Multiplying or dividing by a positive number preserves order
If a < b and c is positive, then ac < bc.
Worked example
Solve 3x ≥ 18. Divide by positive 3: x ≥ 6.
The inequality direction stays the same because division is by a positive number.
7. Multiplying or dividing by a negative number reverses order
This is the most important special rule in elementary inequality solving. If a < b, then multiplying both by −1 gives −a > −b.
Why?
Take 2 < 5. Multiply both sides by −1: −2 and −5. On the number line, −2 is greater than −5. The order reverses because reflection through zero swaps left and right.
Worked example
Solve −4x < 20. Divide by −4 and reverse the sign: x > −5.
Check x = 0: −4(0) = 0 < 20, so 0 should be included. x > −5 does include 0.
8. Do not reverse the sign for ordinary subtraction
Subtracting 3 from both sides of x + 3 > 7 gives x > 4. No reversal is needed.
The reversal is tied specifically to multiplying or dividing both sides by a negative quantity.
Common error
A student sees “moving −3 across” and reverses the inequality. That mixes an informal transposition slogan with the actual algebraic operation. Write the same operation on both sides instead.
9. Two-step inequalities follow the same preservation logic
Worked example
Solve 5x − 7 ≤ 18.
Add 7: 5x ≤ 25. Divide by positive 5: x ≤ 5.
Negative coefficient example
Solve 7 − 3x > 16.
Subtract 7: −3x > 9. Divide by −3 and reverse: x < −3.
10. Expand brackets before isolating the variable when helpful
Solve 2(3x − 1) ≤ 10.
Expand: 6x − 2 ≤ 10. Add 2: 6x ≤ 12. Divide by 6: x ≤ 2.
Alternative legal routes are possible. The key is that each step preserves the solution set.
11. Variables on both sides need collection
Solve 5x + 2 > 2x + 11.
Subtract 2x: 3x + 2 > 11. Subtract 2: 3x > 9. Divide by 3: x > 3.
Check near the boundary
At x = 3, both sides equal 17, so the strict inequality is false. At x = 4, 22 > 19 is true.
12. Compound inequalities describe an interval
The statement 2 < x ≤ 7 means both x > 2 and x ≤ 7. The solution lies between the two boundaries, excluding 2 and including 7.
Worked example
Write “a temperature is above −3°C and at most 5°C” as −3 < T ≤ 5.
This compact notation is useful when one variable must satisfy two conditions simultaneously.
13. Solve a compound inequality across all three parts
Solve −4 ≤ 2x + 2 < 10.
Subtract 2 from all three parts: −6 ≤ 2x < 8.
Divide all three parts by positive 2: −3 ≤ x < 4.
Negative division extension
If dividing all parts by a negative number, every inequality direction reverses. Write the result carefully in increasing left-to-right order.
14. Context can restrict an inequality further
Suppose a box can hold at most 50 whole items, and n represents the number of items. The mathematical constraint is n ≤ 50, but context also requires n to be a non-negative whole number.
The real solution set is therefore not every real number below 50.
Whole-number interpretation
If a calculation gives n > 6.2 and n counts complete buses needed, the smallest feasible whole number is 7. If instead n counts people who may choose to participate, the interpretation may differ.
15. Inequalities model budgets and capacity
An invented activity costs a fixed $12 plus $4 per participant. The budget is at most $60. Let n be the number of participants.
12 + 4n ≤ 60. Then 4n ≤ 48, so n ≤ 12.
If n is a non-negative whole number, at most 12 participants fit the budget under this model.
Check the boundary
For n = 12, cost = $60, allowed because the condition is “at most”. For n = 13, cost = $64, not allowed.
16. Some inequalities have all or no solutions
Consider 2x + 3 < 2x + 8. Subtract 2x: 3 < 8, which is always true. Therefore every real x satisfies the inequality.
Now consider 2x + 9 < 2x + 4. Subtract 2x: 9 < 4, always false. There are no real solutions.
Do not invent x from a vanished variable
If the variable cancels, inspect the resulting statement. It tells whether all values or no values satisfy the original inequality.
17. Inequality graphs are checks, not decorations
After solving x ≥ −2, graph the solution and test a value such as 0. The graph should include 0. Test −3; it should be excluded.
A graph can reveal that the algebraic symbol and shaded direction disagree.
18. Common inequality errors
| Error | Likely issue | Repair prompt |
|---|---|---|
| Reverses sign after adding a negative number | Rule overgeneralised | Did you multiply or divide by a negative? |
| Does not reverse after dividing by −3 | Order reflection missed | What happens to 2 < 5 after multiplying by −1? |
| Uses filled circle for x < 4 | Boundary inclusion confused | Is 4 itself a solution? |
| Writes x < 2 for “at least 2” | Language-symbol mapping weak | Does the boundary value belong? |
| Reports 6.2 buses | Context domain ignored | What values can the variable physically take? |
19. Practice laboratory
- Write “x is at least 7” using an inequality.
- Write “y is less than −2” using an inequality.
- Solve x + 9 > 14.
- Solve 4x ≤ 28.
- Solve −5x > 20.
- Solve 3x − 8 ≥ 13.
- Solve 9 − 2x < 15.
- Solve 4x + 3 > 2x + 11.
- Solve 2(3x + 1) ≤ 20.
- Solve −5 ≤ x + 2 < 8.
- Solve −6 ≤ 2x − 2 ≤ 10.
- State whether x = 4 belongs to x < 4.
- An event venue holds at most 120 people. Write the inequality for whole-number attendance n.
- A service costs $15 plus $6 per unit, with a budget of at most $75. Find the greatest whole number of units.
- Determine the solution set of 3x + 5 < 3x + 9.
- Determine the solution set of 2x + 7 ≥ 2x + 12.
20. Explained answers
1. x ≥ 7.
2. y < −2.
3. x > 5.
4. x ≤ 7.
5. Divide by −5 and reverse: x < −4.
6. 3x ≥ 21, so x ≥ 7.
7. −2x < 6. Divide by −2 and reverse: x > −3.
8. 2x > 8, so x > 4.
9. 6x + 2 ≤ 20, 6x ≤ 18, so x ≤ 3.
10. Subtract 2: −7 ≤ x < 6.
11. Add 2: −4 ≤ 2x ≤ 12. Divide by 2: −2 ≤ x ≤ 6.
12. No. The inequality is strict.
13. 0 ≤ n ≤ 120 with n a whole number, if zero attendance is permitted.
14. 15 + 6u ≤ 75 gives 6u ≤ 60, so u ≤ 10. Greatest whole number: 10.
15. 5 < 9 is always true, so all real x.
16. 7 ≥ 12 is false, so no real solutions.
21. Complete mixed problem
Problem: A fictional delivery has a fixed mass of 8 kg plus 2.5 kg per package. A lift allows at most 58 kg. How many whole packages can be carried under the model?
Let n be the number of packages. 8 + 2.5n ≤ 58.
2.5n ≤ 50, so n ≤ 20. Since n is a non-negative whole number, the maximum is 20 packages.
Boundary check: 8 + 2.5(20) = 58, allowed. For 21 packages, mass is 60.5 kg, which exceeds the limit.
22. Teaching inequalities as preserved order
Begin with true numerical statements such as 2 < 5. Perform the same addition, subtraction, positive multiplication and negative multiplication on both sides. Let the learner observe which operations preserve the direction and which reverse it.
Then move to variables. The rule now has a reason rather than being an isolated memory instruction.
Use boundary tests
After solving, test the boundary and one value on each side. This is especially useful for strict versus inclusive inequalities.
23. Questions students often ask
Why does the sign flip with a negative?
Because multiplying by a negative reflects positions through zero, reversing left-right order.
Do I flip when I subtract a negative number?
No. Adding or subtracting the same quantity from both sides preserves order. The flip occurs when multiplying or dividing by a negative quantity.
Can an inequality have infinitely many answers?
Yes. x > 3 describes infinitely many real values.
Can there be no answers?
Yes. An inequality can reduce to a false statement such as 9 < 4.
What is the fastest check?
Test the boundary and a simple value in the proposed solution region.
24. Return path
Inequalities extend equation reasoning from one equality to a region of possible values. Revisit Equations and Equality when algebraic transformations are unstable, and Numbers and Number Lines when negative-order reasoning is weak.
Continue to Algebraic Formulae, Substitution and Rearrangement for symbolic relationships with several variables.
Sources and learning boundaries
Official curriculum reference: MOE Secondary Syllabus Directory. Exact treatment and sequencing of inequalities vary by subject level and school, so use this as a learning companion rather than a pacing claim.
The budget, lift and capacity examples are independently constructed mathematical models.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Read the order, preserve it under valid operations, test the boundary, respect the domain and return the solution set to the context.
