This guide has everything you need to know on how to represent inequalities on a number line. Once you’ve mastered this, you may want to move on to solving harder linear inequalities.
Step 1: Knowing the Notation
Since solutions to inequalities contain a range of values rather than specific values, they can be represented on a number line. Sometimes, you will be asked to represent a solved inequality, such as 𝑥 < 8. You will sometimes need to solve an inequality first before representing it, such as 𝑥 + 3 < 8.
When solved, draw a circle at any limits. If the inequality symbol for this limit contains an ‘equal to’ inequality sign (≥ or ≤), then shade the circle in. If not, then leave it open (unshaded). Should you find yourself with a single limit, draw an arrow away from the circle to the end of the number line matching the direction of the inequality.
If you have an upper and a lower limit, connect the two circles with a straight line.
Step 2: Drawing the Inequality
Represent the inequality 2 < 𝑥 on a number line.
The inequality means 2 is less than x, or it could be rewritten as x > 2, x is greater than 2. We draw an open circle at 2 and an arrow towards the greater end of the number line.
Represent the inequality -3 ≤ 𝑥 < 4 on a number line.
The inequality means x is greater than or equal to -3 and less than 4, so we draw a solid circle at -3 and an open circle at 4 and then join them with a straight line.
Step 3: Listing the Integer Solutions
List the integer values that satisfy the inequality -3 ≤ 𝑥 < 4.
You can draw a number line to help with this if you need it (as we have for this inequality in Example 2), but it is just asking which whole numbers fall between the limits. Most mistakes will be made at the limits, so look carefully at the symbols to see if you need to include them or not.
Integers that satisfy this inequality are: -3, -2, -1, 0, 1, 2, 3.
4 is not included as the upper limit does not have an ‘equal to’ line on its inequality symbol.
List the integer values of 𝑥 that satisfy the inequality -7 ≤ 3𝑥 – 1 < 8.
To find the limits, we need to solve this inequality so we have the variable, x, alone. We can do this by balancing:
|-7 ≤ 3𝑥 – 1 < 8|
|+ 1||+ 1||+ 1|
|-6 ≤ 3𝑥 < 9|
|÷ 3||÷ 3||÷ 3|
|-2 ≤ 𝑥 < 3|
Integers that satisfy this inequality are: -2, -1, 0, 1, 2.
Step 4: Testing Your Knowledge
Now that you know how to represent inequalities on a number line, you’re probably ready for more revision – you can find more of our blogs here! You can also subscribe to Beyond for access to thousands of secondary teaching resources. You can sign up for a free account here and take a look around at our free resources before you subscribe too.