What you need to know
Before learning about how to display inequalities on a number line, you firstly need to know the following symbols and their definitions :
- > means “greater than”,
- \geq means “greater than or equal to”,
- < means “less than”,
- \leq means “less than or equal to”.
For example, the inequality 2<5 means “2 is less than 5”, which is obviously true. This isn’t all that interesting, really. The more interesting inequalities you’ll see are like
which means that “x is less than or equal to 8”. So, x might be 7 or -10 or 2.34 or \pi or… the list would go on forever, literally. It is called a range of values and is best expressed by x \leq 8 (to save us writing forever).
Note: the inequality x\leq 8 includes the value 8 as a possibility for x, whilst the inequality x<8 does not. We call \leq and \geq inclusive inequalities, and we call < and > strict inequalities.
Inequalities on a Number Line
A common way to display an inequality is using a number line. On an inequality number line, x\leq 8 looks like:
The circle tells us where the end of the inequality is. For an inclusive inequality, use a closed circle like \bullet, and for a strict inequality, use an open circle like \circ. Then, the arrow is used to explain whether to include all the numbers below or all the numbers above where the circle is placed.
Example 1: Inequalities on a Number Line
Display the inequality 3 \leq x<10 on a number line.
This inequality looks like two inequalities put together – and that’s exactly what it is. The first part is 3\leq x, which means that “3 is less than or equal to x” – if you think about what’s being said here, then this is the same thing as saying x \geq 3. Then, the second part is x<10, which means “x is less than 10”.
On a number line the inequality looks like
One part of the inequality was inclusive and the other was strict, so we have one open circle and one closed circle. As you can see, there is no need for an arrow in this situation, since the range of values is enclosed between 3 and 10 and does not get any bigger or smaller.
Example 2: Inequalities on a Number Line
Display the inequality 5 <x \leq10 on a number line.
Similar to the previous example but this time the open and closed circles are the other way round.
1) Display the inequality -1\geq x on a number line.
The inequality, -1\geq x, will require a closed circle at 3 and an arrow pointing right.
2) Display the inequality x \le 4 on a number line.
The inequality, x \le 4, will require an open circle at 4 and an arrow pointing left.
3) Display the inequalities y>3 and y<-2 on the same number line.
The first inequality, y>3, will require an open circle at 3 and an arrow pointing right.
The other inequality, y<-2, will require an open circle at -2 and an arrow pointing left.
4) Display the inequality -2\leq x \leq8 on a number line.
The lower bound, -2\leq x, will require a closed circle at x=-2
The upper bound x \leq8, will require an closed circle at x=8
5) In order to be profitable, a bus company requires a certain number of passengers for each tour; however, the bus can only hold so many seated passengers.
Write an inequality for a bus, b, if the company requires more than 6 passengers to be profitable but can only hold 54 passengers and display this on a number line.
Forming the correct inequality 6\le b \leq 54 and displaying with an open circle for representing the strict inequality (6) and a closed circle representing the non-strict inequality (54).
Worksheets and Exam Questions
Inequalities On a Number Line and Solving Inequalities Q1GCSE MATHS
Inequalities On a Number Line and Solving Inequalities Q2GCSE MATHS
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