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1. Learning Objectives
- Understand the basic properties of linear inequalities.
- Solve inequalities and represent solutions on a number line.
- Use interval notation (e.g., [a, b) or (x, ∞)) correctly.
2. Revision Notes
⚠️ IMPORTANT RULE: If you multiply or divide both sides by a NEGATIVE number, you MUST reverse (flip) the inequality sign.
Interval Symbols:
- ( ) Parentheses: "Open" - endpoint NOT included (< or >).
- [ ] Brackets: "Closed" - endpoint IS included (≤ or ≥).
Click the button below each question to reveal the explanation.
1. Solve for x: x + 7 > 15
Subtract 7 from both sides: x > 8. Ans: x > 8
2. Solve: 3x ≤ 21
Divide by 3: x ≤ 7. Ans: x ≤ 7
3. Solve: -2x < 10
Divide by -2 and FLIP the sign: x > -5. Ans: x > -5
4. Solve: 5 - x ≥ 2
-x ≥ -3. Divide by -1 and flip: x ≤ 3. Ans: x ≤ 3
5. Write "x is greater than 4" in interval notation.
Starting at 4 (not included) to infinity. Ans: (4, ∞)
6. Write x ≤ -2 in interval notation.
From negative infinity to -2 (included). Ans: (-∞, -2]
7. Solve: 2(x + 1) < 8
2x + 2 < 8 → 2x < 6 → x < 3. Ans: x < 3
8. Solve: x/4 + 2 > 5
x/4 > 3 → Multiply by 4: x > 12. Ans: x > 12
9. Solve: 7x - 3 ≤ 5x + 7
2x - 3 ≤ 7 → 2x ≤ 10 → x ≤ 5. Ans: x ≤ 5
10. Does multiplying an inequality by 0 change it?
You cannot multiply or divide by 0 in an inequality as it makes it undefined. Ans: Undefined
11. Solve: -5x ≥ -25
Divide by -5 and flip: x ≤ 5. Ans: x ≤ 5
12. Represent x > 0 in set-builder notation.
Written as {x | x ∈ R, x > 0}. Ans: {x | x > 0}
13. Solve: 3 - 2x < 11
-2x < 8 → Divide by -2 and flip: x > -4. Ans: x > -4
14. What circle is used for ≥ on a number line?
Because the endpoint is included. Ans: Solid/Closed Circle
15. Solve: 4x + 10 > 2
4x > -8 → x > -2. Ans: x > -2
16. Write [5, 10) as an inequality.
5 is included, 10 is not. Ans: 5 ≤ x < 10
17. Solve: x/(-3) < 4
Multiply by -3 and flip: x > -12. Ans: x > -12
18. Solve: 2x + 2 ≤ x + 5
Subtract x: x + 2 ≤ 5 → x ≤ 3. Ans: x ≤ 3
19. Is 5 a solution for x < 5?
No, because it is "strictly less than". Ans: No
20. Solve: 0x < 5
0 < 5 is always true. Ans: All Real Numbers