The equations we are most familiar with are equalities, where the left hand side and the right hand side are identical or equal. It is also useful to have a way of expressing a range of values instead of a single specific value which can be achieved by using inequalities. A common way to display inequalities is using a number line.
Inequality Symbols and their Meaning
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”.
We call \leq and \geq inclusive inequalities, and we call < and > strict inequalities. For example x\leq 8 includes the value 8 as a possibility for x, whilst the inequality x<8 does not.
Type 1: Inequalities on a Number Line
The inequality, x\geq -5 means that x can take any value that is bigger than -5, including -5.
We can display this on a number line by drawing a filled in circle at -5 and an arrow pointing to the right hand side indicating the numbers that are greater than -5. This should look like,
When drawing inequalities on a number line it is important to remember that you should use an closed circle, \bullet, for an inclusive inequality, e.g. x\geq-5 and use a open circle, \circ, for a strict inequality, e.g. x\lt2.
Type 2: Inequalities on a Number Line
Inequalities can describe a range of values between an upper and lower limit. For example the inequality -3 \leq x \lt 5 means that x can take any value greater than or equal to -3 but also has to be less than 5.
Here, the first part of the inequality is an inclusive inequality so is drawn with a filled in circle and the second part is a strict inequality so is drawn with a empty circle.
Example: Draw -3 \leq x \lt 5 on the number line below.
Show the inequality -1 \lt x \lt 10 on a number line.
Key points when drawing the number line.
- The circles indicate the limits of the inequality (-1 and 10).
- Both are strict inequalities they should be open circles.
- A connecting line between the two circles indicates the values x can take between the two limits.
Putting it all together the number line should look like,
Display the inequalities x \leq 0 and x \geq 3 on a number line.
This is similar to the previous example but this time,
- Both are inclusive inequalities so should be drawn with closed circles
- x can take any value less than or equal to 0 so an arrow should be drawn to the left hand side of 0
- x can take any value grater or equal to 3 so an arrow should be drawn to the right hand side of 3
Question 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.
Question 2: Display the inequality x \le 4 on a number line.
The inequality, x \le 4, will require an closed circle at 4 and an arrow pointing left.
Question 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.
Question 4: Display the inequality -1\leq x \leq8 on a number line.
The lower bound, -1\leq x, will require a closed circle at x= -1
The upper bound x \leq8, will require an closed circle at x=8
Question 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< 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).
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