Expressions, formulae and equations
Inequalities
Sometimes you do not know a number exactly — only that it is bigger than this, or smaller than that. An inequality says exactly that.
The two symbols
So far a letter has stood for one unknown number. A letter with an INEQUALITY stands for a whole RANGE of numbers — an open interval. < means 'is less than' > means 'is greater than' x > 4 means x can be any number greater than 4. y < 7 means y can be any number less than 7. In both cases the endpoint is excluded: x cannot equal 4, and y cannot equal 7.
Showing it on a number line
Draw an OPEN CIRCLE at the endpoint — open, because that number is not included — and an arrow showing which way the range continues. For x > 4: an open circle at 4 and an arrow pointing RIGHT, because x can be any number above 4, continuing towards positive infinity. For y < 7: an open circle at 7 and an arrow pointing LEFT, continuing towards negative infinity.
Which integers fit?
If x > 4, the integers that work are 5, 6, 7, 8, … — but not 4 itself. If y < 7, the integers that work are 6, 5, 4, 3, … and on into the negatives. There is no smallest one. A useful check: read the symbol pointing at the smaller side. In x > 4 the wide end faces x, so x is the bigger one.
Worked examples
What does x > 4 mean?
- The > symbol means 'is greater than'.
- The endpoint is not included.
Answer: x is any number above 4, but not 4 itself
What does y < 7 mean?
- The < symbol means 'is less than'.
- 7 itself is excluded.
Answer: y is any number below 7
Showing x > 4 on a number line
- Open circle at 4, because 4 is not included.
- Arrow pointing right.
Answer: An open circle at 4 with an arrow to the right
Which integers satisfy x > 4?
- Start just above 4.
- 4 itself does not count.
Answer: 5, 6, 7, 8, … with no largest one
The smallest integer with y < 7
- The integers below 7 are 6, 5, 4, 3, …
- They carry on into the negatives.
Answer: There isn't one — the list has no end
Reading the symbol the right way round
- The wide end faces the bigger side.
- In x > 4 the wide end faces x.
Answer: So x is the bigger one
Where you see this in real life
- A sign reading 'you must be over 12 to ride' is the inequality age > 12 — and a 12-year-old is not allowed on.
- A lift rated for fewer than 8 people is n < 8, so 7 is fine and 8 is not.
Key words
- Inequality
- a statement that one quantity is greater or less than another
- Open interval
- a range that does not include its endpoints
- Positive infinity
- the direction of endlessly increasing numbers
- Negative infinity
- the direction of endlessly decreasing numbers
- Greater than
- bigger than, shown by >
- Less than
- smaller than, shown by <
- Open circle
- a number-line mark showing the endpoint is not included