LINEAR INEQUALITIES
What Is an Inequality?
An inequality is a mathematical statement that compares two expressions and shows that they are not equal but that one is less than, greater than, less than or equal to, or greater than or equal to the other.
The symbols used include:
- “<” meaning “less than”
- “>” meaning “greater than”
- “≤” meaning “less than or equal to”
- “≥” meaning “greater than or equal to”
Example: “x > 5” means x is greater than 5.
Example: “y ≤ 3” means y is less than or equal to 3.
What Is a Linear Inequality?
A linear inequality is like a linear equation except instead of “=” it uses one of the inequality symbols.
It usually involves one variable, and the term with the variable is of first degree (no squares, no products of variables).
Example: 3x − 4 > 2 is a linear inequality.
Example: 2y + 5 ≤ 11 is another.
Why We Learn Linear Inequalities
- Because many real-life situations are not exact equalities but bounds or limits (e.g. you must score at least 50 to pass, or a car must not exceed 80 km/h).
- To help students understand and solve problems where “less than” or “greater than” relationships occur.
- To prepare for graphical representation (using number lines) which comes later.
What Students Should Be Able to Do
- Recognize inequality statements in words and write them in algebraic form.
- Solve simple linear inequalities (adding, subtracting, multiplying, dividing) similarly to equations—with careful attention to signs.
- Understand that when you multiply or divide by a negative number, the direction of the inequality reverses.
- Represent solutions on a number line (open circle for < or >; closed circle for ≤ or ≥).
- Work out word problems that lead to linear inequalities.
Key Rules / Ideas
- Rules like equations: You can add or subtract the same amount to both sides, or multiply or divide by a positive number without changing the direction, just like in equations.
- Flipping the inequality sign when multiplying or dividing by a negative number: For example, if you have −2x > 6, dividing both sides by −2 gives x < −3 (notice the sign flips).
- Graphical representation:
- Use a number line.
- For “>” or “<”, use an open circle (the equality is not included).
- For “≥” or “≤”, use a closed or filled circle.
- Shade the side of the line that satisfies the inequality.
- Check your solution by substituting a test value to see if it makes the inequality true.
Steps to Solve a Linear Inequality
- Rewrite the inequality, simplifying both sides (expand brackets, combine like terms).
- Isolate the variable by moving other terms (using addition/subtraction).
- Divide or multiply by the coefficient of the variable.
- If dividing or multiplying by a negative, reverse the inequality sign.
- Write the solution (e.g. x > 4, or y ≤ −2).
- Graph the solution on a number line (open / closed circle, shading).
- Check with a value to see if it satisfies the original inequality.
Common Mistakes Students Make
- Forgetting to flip the inequality when multiplying or dividing by a negative.
- Treating inequalities exactly like equations without regard for direction.
- Misplacing open vs closed circle on the number line.
- Not checking with a sample value.
- Writing the solution incorrectly or incompletely.
Linear Inequalities — Questions
-
Problem: Solve 3x + 4 > 13.
Steps:
Subtract 4 from both sides: 3x > 13 − 4 = 9.
Divide both sides by 3: x > 9 ÷ 3 = 3.
Solution: x > 3.
Check: choose x = 4 → 3(4) + 4 = 16 > 13 ✓
Number line: open circle at 3, shade to the right.
-
Problem: Solve 2x − 5 ≤ 9.
Steps:
Add 5 to both sides: 2x ≤ 9 + 5 = 14.
Divide by 2: x ≤ 14 ÷ 2 = 7.
Solution: x ≤ 7.
Check: choose x = 7 → 2(7) − 5 = 9 ≤ 9 ✓
Number line: closed dot at 7, shade to the left.
-
Problem: Solve −3x + 6 ≥ 0.
Steps:
Subtract 6: −3x ≥ −6.
Divide by −3 (flip sign): x ≤ (−6) ÷ (−3) = 2.
Solution: x ≤ 2.
Check: choose x = 2 → −3(2) + 6 = 0 ≥ 0 ✓
Number line: closed dot at 2, shade to the left.
-
Problem: Solve x/2 + 3 < 8.
Steps:
Subtract 3: x/2 < 5.
Multiply both sides by 2: x < 10.
Solution: x < 10.
Check: choose x = 9 → 9/2 + 3 = 7.5 < 8 ✓
Number line: open circle at 10, shade to the left.
-
Problem: Solve 4 − 2x > 10.
Steps:
Subtract 4: −2x > 6.
Divide by −2 (flip sign): x < −3.
Solution: x < −3.
Check: choose x = −4 → 4 − 2(−4) = 12 > 10 ✓
Number line: open circle at −3, shade to the left.
-
Problem: Solve 5(x − 1) ≤ 20.
Steps:
Expand: 5x − 5 ≤ 20.
Add 5: 5x ≤ 25.
Divide by 5: x ≤ 5.
Solution: x ≤ 5.
Check: choose x = 5 → 5(5 − 1) = 20 ≤ 20 ✓
Number line: closed dot at 5, shade left.
-
Problem: Solve (1/3)x − 2 ≥ 1.
Steps:
Add 2: (1/3)x ≥ 3.
Multiply by 3: x ≥ 9.
Solution: x ≥ 9.
Check: x = 9 → (1/3)(9) − 2 = 1 ≥ 1 ✓
Number line: closed dot at 9, shade right.
-
Problem: Solve 7x + 3 < 2x + 18.
Steps:
Subtract 2x: 5x + 3 < 18.
Subtract 3: 5x < 15.
Divide by 5: x < 3.
Solution: x < 3.
Check: x = 2 → 7×2 + 3 = 17 < 22 ✓
Number line: open circle at 3, shade left.
-
Problem: Solve −2(x + 4) ≤ 6.
Steps:
Expand: −2x − 8 ≤ 6.
Add 8: −2x ≤ 14.
Divide by −2 (flip sign): x ≥ −7.
Solution: x ≥ −7.
Check: x = −7 → −2(−7 + 4) = 6 ≤ 6 ✓
Number line: closed dot at −7, shade right.
-
Problem: Solve x/4 + 5 ≥ 8.
Steps:
Subtract 5: x/4 ≥ 3.
Multiply by 4: x ≥ 12.
Solution: x ≥ 12.
Check: x = 12 → 12/4 + 5 = 8 ≥ 8 ✓
Number line: closed dot at 12, shade right.
-
Problem: Solve 1 < 2x + 1 ≤ 7.
Steps:
Subtract 1 from all parts: 0 < 2x ≤ 6.
Divide all parts by 2: 0 < x ≤ 3.
Solution: 0 < x ≤ 3.
Check: x = 1 → 1 < 3 ≤ 7 ✓
Number line: open circle at 0, closed at 3, shade between.
-
Problem: (Word Problem) A taxi fare is ₦30 per kilometre. The total fare must not exceed ₦150. Let k = number of kilometres.
Steps:
30k ≤ 150.
Divide by 30: k ≤ 5.
Solution: k ≤ 5. The greatest whole number of kilometres is 5 km.
Check: 30 × 5 = ₦150 ✓
-
Problem: (Word Problem) A school has 40 chairs and buys boxes with 10 chairs per box. If they need at least 120 chairs, how many boxes b must they buy?
Steps:
40 + 10b ≥ 120.
Subtract 40: 10b ≥ 80.
Divide by 10: b ≥ 8.
Solution: They must buy at least 8 boxes.
Check: 40 + 10×8 = 120 ✓
-
Problem: Solve −4x + 1 < 9.
Steps:
Subtract 1: −4x < 8.
Divide by −4 (flip sign): x > −2.
Solution: x > −2.
Check: x = −1 → −4(−1) + 1 = 5 < 9 ✓
Number line: open circle at −2, shade right.
-
Problem: Solve 3 − x/2 ≥ 1.
Steps:
Subtract 3: −x/2 ≥ −2.
Multiply by −1 (flip sign): x/2 ≤ 2.
Multiply by 2: x ≤ 4.
Solution: x ≤ 4.
Check: x = 4 → 3 − 4/2 = 1 ≥ 1 ✓
Number line: closed dot at 4, shade left.
Quick Tips to Help Students
- Always do the same operation to both sides.
- When you multiply or divide by a negative number, flip the inequality sign.
- For “<” or “>” use open circle on a number line; for “≤” or “≥” use a closed dot.
- Check your final answer by substituting one value from the solution set into the original inequality.
Graphical Representation of Solution of Linear Inequalities in One Variable
Meaning of Graphical Representation
When we solve an inequality such as x > 3 or x ≤ 5, we often want to show all possible values of x that satisfy it. A graphical representation simply means showing the solution on a number line instead of only writing it in algebraic form. It helps students to see the range of possible answers visually.
The Number Line
A number line is a straight line on which numbers are arranged in increasing order from left to right. It usually has zero in the middle, positive numbers to the right, and negative numbers to the left.
← −3 −2 −1 0 1 2 3 4 5 →
Symbols and Their Graphical Meanings
| Inequality Symbol | Meaning in Words | Type of Circle/Dot | Direction of Shading |
|---|
| x < a | x is less than a | Open circle at a | Shade to the left |
| x > a | x is greater than a | Open circle at a | Shade to the right |
| x ≤ a | x is less than or equal to a | Closed circle at a | Shade to the left |
| x ≥ a | x is greater or equal to a | Closed circle at a | Shade to the right |
🔹 Open circle (○) → means the number is not included.
🔹 Closed circle (●) → means the number is included.
Steps to Represent a Linear Inequality on a Number Line
- Solve the inequality if it is not already simplified. Example: 3x − 4 ≤ 8 ⇒ x ≤ 4.
- Draw a horizontal number line with suitable scale.
- Mark the critical value (the boundary number, e.g., 4).
- Use open circle (○) if < or > and closed circle (●) if ≤ or ≥.
- Shade or arrow the part of the line showing all values that make the inequality true.
Worked Examples
Example 1: Solve and represent x < 3 on a number line.
Step 1: The solution is already in simple form x < 3.
Step 2: Draw a number line showing values from 0 to 5.
Step 3: Put an open circle at 3 (since it is < not ≤).
Step 4: Shade the left side of 3.
← ●────────○────────→
1 2 3 4
Example 2: Solve and represent x ≥ −2 on a number line.
Step 1: Inequality is x ≥ −2.
Step 2: Draw a number line from −5 to 3.
Step 3: Place a closed circle at −2.
Step 4: Shade the right side of −2.
←──────●════════════⇒
-3 -2 -1 0 1 2
Example 3: Solve and represent −3x + 6 > 0 on a number line.
Step 1: Solve algebraically:
−3x + 6 > 0
Subtract 6: −3x > −6
Divide by −3 (flip sign): x < 2.
Step 2: Draw a number line around 2.
Step 3: Put an open circle at 2.
Step 4: Shade left side of 2.
←════════○────────→
1 2 3 4
Example 4: Solve and represent 2x + 1 ≤ 5 on a number line.
Step 1: Simplify:
Subtract 1: 2x ≤ 4
Divide by 2: x ≤ 2.
Step 2: Draw a number line near 2.
Step 3: Closed circle at 2 (since ≤).
Step 4: Shade to the left.
←══════●────────→
0 1 2 3
Compound Inequalities
A compound inequality has two inequality signs that describe a range of values.
Example: 1 < x ≤ 4 means x is greater than 1 but less than or equal to 4.
Steps:
- Draw a number line from 0 to 5.
- Open circle at 1 (since <).
- Closed circle at 4 (since ≤).
- Shade between 1 and 4.
0 ○════● 5
1 4
Word Problem Example
A taxi can carry at most ₦5000 worth of passengers per trip. Let x represent the amount collected. Write the inequality and represent the solution on a number line.
Step 1: “At most ₦5000” means x ≤ 5000.
Step 2: On a number line, mark 5000.
Step 3: Closed circle at 5000, shade to the left (since the fare must not exceed ₦5000).
Key Points to Remember
- Always solve the inequality first before drawing.
- Remember to flip the inequality sign if you multiply or divide by a negative number.
- Use open circle for < or >; closed circle for ≤ or ≥.
- Shade the direction of the true values: left for smaller values, right for larger values.
- For compound inequalities, shade between the two limits.
Class Activity / Practice Exercise
- Represent x ≥ 3 on a number line.
- Represent x < −1 on a number line.
- Represent −2 ≤ x < 4 on a number line.
- Solve and represent 2x + 5 > 9 on a number line.
- Solve and represent −3x + 6 ≤ 0 on a number line.
Linear Inequalities — Word Problems
Example 1: Taxi fare
A taxi fare is ₦30 per kilometre. Total fare ≤ ₦150. Let k = number of kilometres.
Step 1: 30k ≤ 150
Step 2: k ≤ 150 ÷ 30 = 5
Step 3: graph:
0 1 2 3 4 5
●========================>
Example 2: Buying chairs
A school has 40 chairs and buys boxes with 10 chairs per box. They need at least 120 chairs. Let b = number of boxes.
Step 1: 40 + 10b ≥ 120
Step 2: 10b ≥ 80 → b ≥ 8
Step 3: graph:
0 1 2 3 4 5 6 7 8
●================>
Example 3: Maximum weight of sugar
A shop must not carry more than 50 kg of sugar. Let x = number of sacks.
x ≤ 50
0 10 20 30 40 50
●==============================>
Example 4: Age restriction
Club admits members older than 15 years. Let x = age of member.
x > 15
15 16 17 18 19
○========================>
Example 5: Compound inequality (ages between 12 and 18)
School accepts students ≥12 but <18 years old. Let x = age.
12 ≤ x < 18
12 13 14 15 16 17 18
●==============================○
Example 6: Limited items
Store sells at most 30 pens. Let p = number of pens sold.
p ≤ 30
0 10 20 30
●====================>
Example 7: Minimum height for ride
Children must be ≥120 cm to ride. Let h = height.
h ≥ 120
100 110 120 130 140
●================>
Example 8: Salary restriction
Worker earns >₦20,000/month to get a bonus. Let s = salary.
s > 20000
20000 21000 22000 23000
○==============================>
Example 9: Maximum number of students
Bus carries ≤50 students. Let n = number of students.
n ≤ 50
0 10 20 30 40 50
●==============================>
Example 10: Temperature requirement
Temperature must be below 30°C. Let t = temperature.
t < 30
25 26 27 28 29 30
==============================○
Example 11: Exam marks (compound inequality)
Students pass if marks ≥40 but <70. Let m = marks.
40 ≤ m < 70
40 45 50 55 60 65 70
●==============================○
Example 12: Delivery limit
Truck carries >10 boxes but ≤50 boxes. Let x = boxes.
10 < x ≤ 50
10 15 20 25 30 40 50
○==============================●
Example 13: Minimum score for scholarship
Student must score ≥80 marks. Let s = score.
s ≥ 80
70 75 80 85 90
●================>
Example 14: Maximum bottles purchased
Customer can buy at most 12 bottles. Let b = number of bottles.
b ≤ 12
0 2 4 6 8 10 12
●==============================>
Example 15: Age restriction for competition
Participants must be older than 14. Let x = age.
x > 14
14 15 16 17 18
○========================>
