We'll often need to find all of the numbers that satisfy a given inequality. This is known as solving an inequality
For instance we may be asked to represent all of the numbers that satisfy the inequality:
\[x < 5 \]
In other words we need to state all the numbers for which that inequality is true. There are an infinite amount of numbers
for which this inequality is true. Indeed, so long as \(x\) is a number that is less than \(5\) this inequality will be true. So
\(4.9,\ 4,\ 3,\ 1,\ 0,\ -7,\ -52, \ \dots \) are all numbers that satisfy this inequality ("solutions to the inequality").
Since it isn't possible to write all of the numbers that satisfy this inequality, we often use a graphical representation of the solution. We do so using
the number line.
The representation of the solutions of the inequality \(x<5\) is shown here:
The pink arrow starts at \(5\) and illustrates all of the numbers that satisfy the inequality.
The arrow points in the direction of all of the solutions.
Each of the four inequality symbols is illustrated here.
These
Illustrate the solutions to each of the following inequalities using the number line: