Inequalities (Edexcel GCSE Maths): Revision Notes
Inequalities
Understanding inequality symbols
When working with inequalities, you need to understand what each symbol means. These symbols help you compare numbers and show relationships between them.
The four main inequality symbols are:
- > means "is greater than"
- ≥ means "is greater than or equal to"
- < means "is less than"
- ≤ means "is less than or equal to"
Remember that the symbol always points towards the smaller number, like an arrow pointing to the lesser value.
Error intervals
Error intervals show the range of possible values when a number has been rounded. They use inequalities to express this range clearly.
How error intervals work
When you round a number, the actual value could be anywhere within a specific range. Error intervals help you write this range using inequality notation.
Key examples
Worked Example: Height Measurement
If the London Eye's height is given as 140m (rounded to the nearest 10m), the actual height could be anywhere from 135m up to (but not including) 145m.
Solution:
- This is written as:
- 135m is the smallest possible value (included)
- 145m is the largest possible value (not included)
Worked Example: Decimal Rounding
If a number rounds to 4.7 (to 1 decimal place), find the error interval.
Solution: The original value was between 4.65 and 4.75.
- This is written as:
Worked Example: Two Decimal Places
If a number rounds to 8.64 (to 2 decimal places), find the error interval.
Solution: The error interval is:
Key Rule for Error Intervals:
- Use "" (less than or equal to) for the smallest possible value
- Use "" (less than) for the largest possible value
Inequalities on number lines
Number lines provide a visual way to represent inequalities. They make it easy to see which values are included or excluded.
Circle symbols on number lines
Understanding the circle symbols is essential for reading number line inequalities:
- Open circle (○): The value is not included in the solution
- Closed circle (●): The value is included in the solution
Reading number line inequalities
Worked Example: Open Circle
If you see an open circle at -1 with an arrow pointing right:
- This represents (x is greater than -1)
- The value -1 is not included
Worked Example: Closed Circle
If you see a closed circle at 3 with an arrow pointing left:
- This represents (x is less than or equal to 3)
- The value 3 is included
Compound inequalities on number lines
You can show ranges of values using number lines. For example, means x is greater than -1 but less than or equal to 4. This shows as an open circle at -1 and a closed circle at 4, with a line connecting them.
Finding integer solutions
When asked for integer values that satisfy an inequality:
- Integers are whole numbers (positive, negative, and zero)
- Look carefully at whether boundary values are included or excluded
- For , the integers are: 0, 1, 2, 3, 4
When finding integer solutions, pay special attention to whether boundary values should be included or excluded based on the inequality symbols used.
Exam tips
Exam Strategy for Inequalities:
When working with inequalities in exams:
- Read the inequality symbols carefully - check whether boundary values are included
- For "less than" (<), don't include the boundary value
- For "less than or equal to" (≤), do include the boundary value
- Include zero in your integer solutions when appropriate
- Double-check your number line representations match your written inequalities
Key Points to Remember:
- Inequality symbols always point towards the smaller number
- Open circles mean "not included", closed circles mean "included"
- Error intervals show the range of possible values for rounded numbers
- Use for the minimum value and for the maximum value in error intervals
- Always check whether boundary values should be included when finding integer solutions