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Math · Inequalities
Inequalities: Boundaries, Number Lines, Solve and Check
Move from equations to inequalities by interpreting <, >, ≤ and ≥, finding boundary values, solving two-step inequalities with positive multipliers/divisors, graphing solution sets and checking boundary cases.
The idea
An equation usually asks for value(s) that make two expressions equal. An inequality describes a range of values.
The boundary matters. x < 5 excludes 5, while x ≤ 5 includes 5.
For today's lesson, all solving steps divide or multiply by positive numbers, so the inequality symbol keeps its direction. Reversing the sign when multiplying or dividing by a negative number is a later extension.
Worked examples
Example 1 · Strict boundary
Solve 3x + 4 < 19.
- Subtract 4 from both sides: 3x < 15.
- Divide by positive 3: x < 5.
- Graph an open circle at 5 and shade left.
- Check: x = 4 works; x = 5 does not.
x < 5.
Example 2 · Included boundary
Solve 2n + 7 ≤ 17.
- Subtract 7: 2n ≤ 10.
- Divide by positive 2: n ≤ 5.
- Use a closed circle at 5 and shade left.
n ≤ 5.
Example 3 · Greater than
Solve 5a − 6 > 14.
- Add 6: 5a > 20.
- Divide by positive 5: a > 4.
- Open circle at 4, shade right.
a > 4.
Example 4 · Real constraint
A bag may weigh at most 10 kg.
- Let w be weight in kilograms.
- At most means the boundary is included.
- Write w ≤ 10.
w ≤ 10.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Strict < excludes the boundary, so use an open circle.
Why this fails: At least means the value can equal the boundary or be greater: ≥.
Why this fails: Check the boundary and one value on each side to understand the solution set.
Why this fails: As with equations, preserve the relation by applying the same operation to both sides.
Why this fails: The sign reverses only when multiplying or dividing by a negative quantity.
Foundation
Foundation practice
Read symbols and solve straightforward two-step inequalities.
1. 3x + 4 < 19
2. 2n + 7 ≤ 17
3. 5a − 6 > 14
4. 4m − 3 ≥ 13
5. x ÷ 3 + 2 < 6
6. y ÷ 5 − 1 ≤ 3
7. 2k + 9 > 9
8. 6t − 12 ≥ 0
9. Does x = 4 satisfy x < 5?
10. Does x = 5 satisfy x < 5?
Core
Core practice
Include negative boundary values and boundary reasoning.
11. 7x + 1 ≤ 29
12. 9y − 8 > 28
13. 3a + 12 ≥ 3
14. 4b − 5 < −1
15. c ÷ 6 + 4 ≤ 9
16. d ÷ 7 − 2 > 3
17. 2p − 15 ≥ −5
18. 5q + 12 < 7
19. What is the boundary value in 3x + 4 < 19?
20. Why is x = 5 excluded from x < 5?
Reasoning
Reasoning practice
Translate constraints and interpret number-line boundaries.
21. A ride has a height rule h ≥ 120 cm. What does it mean?
22. A bag may weigh w ≤ 10 kg. Is 10 kg allowed?
23. A game requires score s > 50. Is 50 enough?
24. You have $30 and tickets cost $6 each plus a $6 fee. Write an inequality for at most $30.
25. Solve 6t + 6 ≤ 30.
26. A shelf holds fewer than 40 books. Write an inequality.
27. A temperature must be at least −5°C. Write an inequality.
28. Which includes 8: x < 8 or x ≤ 8?
29. What graph symbol shows an included boundary?
30. What graph symbol shows an excluded boundary?
Stretch
Stretch practice
Test boundaries, translate constraints and explain why a value belongs or does not belong.
31. For x ≥ 7, are 6, 7 and 8 solutions?
32. Describe the graph of n > −2.
33. Describe the graph of m ≤ 3.
34. A pool requires swimmers to be at least 120 cm tall. Write the inequality.
35. A room may contain no more than 25 people. Write the inequality.
36. Solve 4x + 2 ≤ 18, then test the boundary.
37. A student graphs x < 6 with a closed circle at 6. What should change?
38. What extra rule will be needed later when solving −2x > 8?
Answer key
- 1. x < 5
Subtract 4, then divide by 3. - 2. n ≤ 5
Subtract 7, then divide by 2. - 3. a > 4
Add 6, then divide by 5. - 4. m ≥ 4
Add 3, then divide by 4. - 5. x < 12
Subtract 2, then multiply by 3. - 6. y ≤ 20
Add 1, then multiply by 5. - 7. k > 0
Subtract 9, then divide by 2. - 8. t ≥ 2
Add 12, then divide by 6. - 9. Yes
4 is less than 5. - 10. No
A strict < inequality does not include the boundary. - 11. x ≤ 4
Subtract 1 and divide by 7. - 12. y > 4
Add 8 and divide by 9. - 13. a ≥ −3
Subtract 12 and divide by 3. - 14. b < 1
Add 5 and divide by 4. - 15. c ≤ 30
Subtract 4 and multiply by 6. - 16. d > 35
Add 2 and multiply by 7. - 17. p ≥ 5
Add 15 and divide by 2. - 18. q < −1
Subtract 12 and divide by 5. - 19. 5
The related equation 3x + 4 = 19 has solution 5. - 20. Because < means strictly less than.
Only ≤ includes the boundary. - 21. A rider must be at least 120 cm tall.
The boundary is included because of ≥. - 22. Yes.
≤ includes the boundary. - 23. No.
The score must be strictly greater than 50. - 24. 6t + 6 ≤ 30
Total cost cannot exceed 30. - 25. t ≤ 4
Subtract 6, then divide by 6. - 26. b < 40
Fewer than means strictly less than. - 27. T ≥ −5
At least includes the boundary. - 28. x ≤ 8
≤ includes equality. - 29. A closed/filled circle.
Closed means equality is allowed. - 30. An open circle.
Open means the boundary is not included. - 31. 7 and 8 are solutions; 6 is not.
The boundary 7 is included, and values greater than 7 also satisfy the inequality. - 32. Open circle at −2 and shade to the right.
> excludes the boundary and includes larger values. - 33. Closed circle at 3 and shade to the left.
≤ includes the boundary and all smaller values. - 34. h ≥ 120
At least means 120 is allowed and larger heights are allowed. - 35. p ≤ 25
No more than means the maximum is 25, including 25. - 36. x ≤ 4; x = 4 gives 18 ≤ 18, so the boundary works.
Subtract 2, divide by positive 4, then substitute x = 4. - 37. Use an open circle at 6.
A strict inequality excludes the boundary value. - 38. Multiplying or dividing an inequality by a negative number reverses the inequality sign.
Today's lesson avoids negative divisors, but this is the key extension rule.