Inequalities on a Number Line
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What the inequality signs mean, how to show an inequality on a number line with open and filled circles, how to read one from a number line, and how to list the integers that fit.
What an inequality means
An inequality says that one thing is bigger or smaller than another. 7 > 3 means 7 is greater than 3, and 3 < 7 means 3 is less than 7. The pointed end of the sign always points to the smaller number.
- < means is less than
- > means is greater than
- \leq means is less than or equal to
- \geq means is greater than or equal to
With a letter, an inequality describes lots of numbers at once. x > 3 means x can be any number greater than 3, such as 3.5, 4 or 100, but not 3 itself. x \geq 3 allows 3 as well.
You can turn an inequality round, as long as the sign still points to the smaller side. x > 3 and 3 < x say the same thing.
In the exam you are usually asked to write down the inequality shown on a number line, to show an inequality on a number line, or to write down the integers that fit an inequality, or the largest one, for 1 or 2 marks each. These often come as parts (a) and (b) of one question. Now and then you are shown someone else's number line and asked to write down the mistakes.
Showing an inequality on a number line
On a number line, a circle marks the number where the inequality starts, and an arrow shows which way the other values go.
- An open circle (left empty) is for < or >. The number is not included.
- A filled circle (coloured in) is for \leq or \geq. The number is included.
One way to remember it: the signs with an extra line, \leq and \geq, use extra ink, and so does their circle.
Greater than means bigger numbers, which are to the right, so for > or \geq the arrow points right. Less than means smaller numbers, so for < or \leq the arrow points left.
Worked example 1: Reading an inequality from a number line
The number line shows a filled circle at -3 with an arrow pointing to the right. Write down the inequality shown on the number line.
- Here is the number line.
- The arrow points right, so x is greater than -3
- The circle is filled, so -3 is included. The sign is \geq, not >
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Your turn. Write down the inequality.
x is greater than or equal to -3: x \geq -3
- Answer: x \geq -3
Integers that fit an inequality
An integer is a whole number. Integers can be positive, negative or zero: \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots
If x is an integer and x \leq 4, the largest value x can be is 4, because 4 is included. If x < 4, then 4 is not allowed, so the largest integer is 3
The end of an inequality does not have to be a whole number. For x < 1.5 the open circle goes halfway between 1 and 2, and the largest integer is 1
Take care with negative numbers. Numbers get smaller as you go left, so -3 is less than -1. The integers greater than -3 are -2, -1, 0, 1, \ldots
Worked example 2: Showing an inequality and finding the largest integer
(a) Show the inequality x < -1 on a number line. (b) x is an integer. Write down the largest possible value of x.
- (a) The sign is <, so -1 is not included. Draw an open circle at -1
- Less than means smaller numbers, so draw an arrow from the circle to the left.
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Your turn. (b) Write down the largest integer that is less than -1
-1 is not included, and the next integer to its left is -2
- Check: -2 < -1 is true. But -1 is not less than -1, and 0 is greater than -1
- Answer: (a) an open circle at -1 with an arrow to the left (b) -2
Inequalities between two numbers
Sometimes x is between two numbers. 1 < x \leq 4 means x is greater than 1 and less than or equal to 4. Read each sign from x: 1 < x says x is greater than 1, and x \leq 4 says x is at most 4
On a number line, put a circle at each end, open or filled to match the sign next to that number. Then join the two circles with one straight line. There are no arrows, because the values stop at both ends.
When you write an inequality like this, put the smaller number on the left, as it is on the number line, and the letter in the middle, as in 0 \leq x < 6
Worked example 3: Reading an inequality between two numbers
The number line shows an open circle at -2 and a filled circle at 1, joined by a line. Write down the inequality shown on the number line.
- Here is the number line. The values of x are between -2 and 1
- The circle at -2 is open, so -2 is not included. Use < at that end.
- The circle at 1 is filled, so 1 is included. Use \leq at that end.
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Your turn. Write down the inequality, with the smaller number on the left.
Put x in the middle: -2 < x \leq 1
- Answer: -2 < x \leq 1
Listing the integers between two numbers
To list the integers, look at each end. At the left end, start at the number itself if the sign is \leq, or at the next integer up if the sign is <. At the right end, stop at the number itself if the sign is \leq, or at the integer before it if the sign is <.
For -2 < n \leq 3: -2 is not included, so start at -1. 3 is included, so stop at 3. The integers are -1, 0, 1, 2, 3
Write every integer in order, from one end to the other. Don't forget 0: it is an integer.
Worked example 4: Showing an inequality and listing the integers
(a) Show the inequality -1 \leq y < 4 on a number line. (b) y is an integer. Write down all the possible values of y.
- (a) The sign next to -1 is \leq, so -1 is included. Draw a filled circle at -1
- The sign next to 4 is <, so 4 is not included. Draw an open circle at 4
- Join the two circles with one straight line.
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Your turn. (b) Write down all the integers y can be. Separate them with commas.
Start at -1, which is included, and stop before 4, which is not: -1, 0, 1, 2, 3
- Answer: (a) a filled circle at -1 and an open circle at 4, joined by a line (b) -1, 0, 1, 2, 3
Worked example 5: Integers from a number line
The number line shows an inequality for n: a filled circle at -7 and an open circle at -3, joined by a line. n is an integer. (a) Write down the largest possible value of n. (b) Write down all the possible values of n.
- Here is the number line.
- The circle at -7 is filled, so -7 is included. The circle at -3 is open, so -3 is not included.
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Your turn. (a) Write down the largest possible value of n
-3 is not included, so the largest integer is the one just to its left: -4
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Your turn. (b) Write down all the possible values of n. Separate them with commas.
Start at -7, which is included, and stop at -4: -7, -6, -5, -4
- Answer: (a) -4 (b) -7, -6, -5, -4
Worked example 6: Spotting the mistakes in a number line
Priya is asked to show the inequality -1 \leq x < 3 on a number line. Here is her answer. Write down two mistakes Priya has made.
- Here is Priya's answer.
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Your turn. Write down the inequality that Priya's number line shows.
An open circle at -1 and an arrow to the right show x > -1
- Mistake 1: the sign next to -1 is \leq, so -1 is included. The circle at -1 should be filled.
- Mistake 2: the values stop before 3, so the line should end at an open circle at 3, with no arrow.
- Say which circle and which number each time. Just writing that the circle is wrong is too vague to score.
- Answer: the circle at -1 should be filled, and the line should stop at an open circle at 3 instead of ending in an arrow.
Common mistakes
- Mixing up open and filled circles. Use an open circle for < or >, and a filled circle for \leq or \geq.
- Drawing arrows on an inequality between two numbers. Join the two circles with one line and stop at both circles.
- Using crosses, or the number line itself, instead of circles and a line. Draw your own circles at the ends and your own line between them, just above the number line.
- Leaving out the letter. -2 < 1 is not an inequality for x, and nor is a list of integers. Put the letter in the middle: -2 < x \leq 1
- Including an end that is not allowed. If x < 4 and x is an integer, the largest value is 3, not 4, and not a decimal such as 3.9
- Missing out zero or the negative numbers. Write every integer in order from the left end to the right end.
More on this topic: Inequalities on a Number Line worksheet with full solutions