A22: Inequalities
Solve linear inequalities; represent solutions on number line and using set notation; quadratic inequalities (Higher)
📋 Key Concepts
| Symbol | Meaning |
|---|---|
| < | Less than |
| > | Greater than |
| ≤ | Less than or equal to |
| ≥ | Greater than or equal to |
Key rule: Solve like equations, BUT when multiplying or dividing by a negative number, reverse the inequality sign.
📝 Solving Linear Inequalities
Example 1
Solve: 3x + 2 > 11
Solution:
Subtract 2: 3x > 9
Divide by 3: x > 3
Example 2
Solve: 5x - 3 ≤ 2x + 9
Solution:
Subtract 2x: 3x - 3 ≤ 9
Add 3: 3x ≤ 12
Divide by 3: x ≤ 4
Example 3
Solve: -2x < 8
Solution:
Divide by -2 and reverse sign: x > -4
Remember: when dividing by negative, flip the sign!
📝 Representing on a Number Line
Notation:
- < or >: Open circle (not including)
- ≤ or ≥: Filled circle (including)
Example 4
Represent x > 2 on a number line.
Solution: Open circle at 2, arrow pointing right (towards greater values).
Example 5
Represent -3 ≤ x < 5 on a number line.
Solution: Filled circle at -3, open circle at 5, line between them.
📝 Integer Solutions
Example 6
List the integer solutions of -2 ≤ x < 4
Solution:
x can be -2, -1, 0, 1, 2, 3
(Note: x cannot be 4 because of the strict inequality)
📝 Set Notation
Set notation:
- {x : x > 3} means "the set of all x such that x > 3"
- {x : -2 ≤ x < 5} means x is between -2 and 5
Example 7
Write x ≥ 1 in set notation.
Solution: {x : x ≥ 1}
📝 Quadratic Inequalities (Higher)
Method:
- Rearrange to ax² + bx + c < 0 (or >, ≤, ≥)
- Find where y = ax² + bx + c = 0 (the roots)
- Sketch the graph
- Identify which region satisfies the inequality
Example 8
Solve: x² - 3x - 4 > 0
Solution:
Step 1: Find roots: x² - 3x - 4 = 0
(x - 4)(x + 1) = 0
x = 4 or x = -1
Step 2: Sketch parabola (opens upward, a > 0)
Step 3: We want where y > 0 (above x-axis)
Answer: x < -1 or x > 4
Example 9
Solve: x² + 2x - 15 ≤ 0
Solution:
Roots: (x + 5)(x - 3) = 0
x = -5 or x = 3
Parabola opens upward. We want y ≤ 0 (below or on x-axis)
Answer: -5 ≤ x ≤ 3
❓ Practice Questions
Q1: Solve: 4x - 7 > 13
Q2: Solve: 2x + 5 ≤ x + 8
Q3: Solve: -3x > 12
Q4: List the integer values satisfying -3 < x ≤ 2
Q5: Solve: x² - 9 > 0
Q6: Solve: x² + 5x + 6 < 0
✅ Answers
- x > 5
- x ≤ 3
- x < -4
- -2, -1, 0, 1, 2
- x < -3 or x > 3
- -3 < x < -2
🎯 Exam Tips
- When multiplying/dividing by negative: flip the sign!
- Open circle for < or >, filled for ≤ or ≥
- For quadratics: sketch the graph to see the solution region
- Above x-axis = positive, below = negative
- Check a value in each region to verify
🧠 Problem-Solving Strategies
Problem-Solving
Solve inequalities like equations, but FLIP the sign when multiplying or dividing by a negative. For quadratic inequalities, sketch the parabola first: identify roots, then decide which regions satisfy the inequality. Above x-axis = positive, below = negative.
Multi-Step Problem
The length of a rectangle is (x + 5) cm and width is (x - 2) cm. The area must be greater than 30 cm². Find the range of x.
Solution:
(x + 5)(x - 2) > 30 → x² + 3x - 10 > 30 → x² + 3x - 40 > 0
Factorise: (x + 8)(x - 5) > 0
Parabola opens up. We need positive region: x < -8 or x > 5
But x - 2 > 0 (width must be positive), so x > 2. Combined: x > 5 cm
⚠️ Common Errors
Watch Out!
1. Wrong: Dividing -4x > 12 by -4 gives x > -3 Correct: When dividing by negative, FLIP the sign: x < -3
2. Wrong: Solving x² > 4 as x > 2 only Correct: x² > 4 means x > 2 OR x < -2 (both directions from zero)
3. Wrong: Writing -3 < x < 5 as {x: x < 5 and x > -3} Correct: Use proper set notation {x : -3 < x < 5} or separate inequalities connected by "and"
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A company's profit is P = -x² + 12x - 20 thousand pounds, where x is the price in pounds. (a) Find the prices that give zero profit. (b) Find the price range for positive profit. (c) What price maximises profit?
(a) -x² + 12x - 20 = 0 → x² - 12x + 20 = 0 → (x - 2)(x - 10) = 0 → x = 2 or x = 10
(b) Parabola opens downward (negative x²). Positive profit between roots: 2 < x < 10
(c) Maximum at midpoint of roots: x = 6. P = -36 + 72 - 20 = 16 thousand pounds (£16,000)
Mark scheme: (a) 2 marks. (b) 2 marks for correct inequality with reasoning. (c) 2 marks.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A theme park ride has a height restriction: riders must be at least 120 cm tall but under 200 cm.
(a) Write this as an inequality using h for height.
(b) A child is 1.3 m tall. Can they ride?
(c) The park changes the rule: riders must be at least 140 cm OR accompanied by an adult. Explain why this changes the set of allowed riders.
Answers: (a) 120 ≤ h < 200 (cm). (b) 1.3 m = 130 cm. Yes, 120 ≤ 130 < 200. (c) The "OR" means children under 140 cm can ride if accompanied, expanding the set of riders. Previously they were excluded entirely.
📝 Exam Questions by Topic
🎬 Video Resources
Ready to ace your GCSE Mathematics exams?
Get the best revision books and guides to boost your grades.