Forming linear inequalities
Inequalities are the relationships between two expressions which are not equal to one another. Here are some symbols for inequalities:
| Symbol | Meaning |
| \({\textless}\) | \({y}~{\textless}~{x}\) reads as ‘\({x}\) is greater than \({y}\)’ or ‘\({y}\) is less than \({x}\)’ |
| \({\textgreater}\) | \({7}~{\textgreater}~{x} \) reads as '\({7}\) is greater than \({x}\)' or '\({x}\) is less than \({7}\)'. |
| \({\leq}\) | \({x}~{\leq}~{-4}\) reads as ‘\({x}\) is less than or equal to \({-4}\)’ or ‘\({-4}\) is greater than or equal to \({x}\)’ |
| \({\geq}\) | \({z}~{\geq}~{13}\) reads as ‘\({z}\) is greater than or equal to \({13}\)’ or ‘\({13}\) is less than or equal to \({z}\)’ |
| Symbol | \({\textless}\) |
|---|---|
| Meaning | \({y}~{\textless}~{x}\) reads as ‘\({x}\) is greater than \({y}\)’ or ‘\({y}\) is less than \({x}\)’ |
| Symbol | \({\textgreater}\) |
|---|---|
| Meaning | \({7}~{\textgreater}~{x} \) reads as '\({7}\) is greater than \({x}\)' or '\({x}\) is less than \({7}\)'. |
| Symbol | \({\leq}\) |
|---|---|
| Meaning | \({x}~{\leq}~{-4}\) reads as ‘\({x}\) is less than or equal to \({-4}\)’ or ‘\({-4}\) is greater than or equal to \({x}\)’ |
| Symbol | \({\geq}\) |
|---|---|
| Meaning | \({z}~{\geq}~{13}\) reads as ‘\({z}\) is greater than or equal to \({13}\)’ or ‘\({13}\) is less than or equal to \({z}\)’ |
Inequalities on a number line
Inequalities can be shown on a number line.
Open circles are used for numbers that are less than or greater than (\({\textless} \) or \({\textgreater}\)).
Closed circles are used for numbers that are less than or equal to and greater than or equal to (\({\leq}\) or \({\geq}\)).
For example, this is the number line for the inequality \({x}~{\geq}~{o}\):
The symbol used is greater than or equal to (\({\geq}\)) so a closed circle must be used at \({0}\). \({x}\) is greater than or equal to \({0}\), so the arrow from the circle must show the numbers that are larger than \({0}\). The arrow head shows that all the numbers above \({3}\) are also included in the inequality.
Example
Show the inequality \({y}~{\textless}~{2}\) on a number line.
Solution
\({y}\) is less than (\({\textless}\)) \({2}\), which means an open circle at \({2}\) must be used. \({y}\) is less than \({2}\), so an arrow below the values of \({2}\) must be drawn in. The arrowhead means that all the numbers less than \({-5}\) are also included in the inequality.
Question
What inequality is shown by this number line?
There is a closed circle at \({-5} \) with the line showing the numbers that are greater than \({-5} \).
This means \({x}~{\geq}~{-5}\).
There is also an open circle at \({4}\), with the numbers less than \({4}\) indicated. This means \({x}~{\textless}~{4} \).
The line between these two points means that \({x}\) satisfies both inequalities, so a double inequality must be created.
Putting \({x}\) in the middle of the two inequalities gives \({-5}~{\leq}~{x}~{\textless}~{4}\).
\({x}\) is greater than or equal to \({-5}\) and \({x}\) is less than \({4}\).
Remember that this means \({x}\) can be any value in this range – including, for example, decimals such as \({2.045}\).
More guides on this topic
- Equations of lines – WJEC
- Equations of curves - Intermediate & Higher tier – WJEC
- Basic algebra – WJEC
- Equations and formulae – WJEC
- Factorising - Intermediate & Higher tier – WJEC
- Quadratic expressions - Intermediate & Higher tier – WJEC
- Sequences – WJEC
- Functions - Higher only – WJEC
- Simultaneous equations - Intermediate & Higher tier – WJEC