Help with solving inequalities... the bigger than and less than signs < and >
The bigger than and less than signs
Inequalities help us to define a range of values for a variable such as x.
To do this we use the bigger than and less than signs. The look like pointers... > and <.
Remember, the biggest number always goes on the open side of the pointer.
The smaller number always goes on the sharp, pointed side of the pointer.
The pointer points to the smaller number.
The open side of the sign is like a big fish's mouth open to take a really big bite.
A good way of working out inequalities is to use a number line. You can get a number line here or you can draw your own.
Note, you can use negative numbers, like -1, in inequalities. You can also use other numbers too such as fractions or decimals.
There are two other signs that look similar, ³ and £. They mean "bigger than or equal to..." and "less than or equal to..." There some examples of how to use these signs below.
This mean that y can be any number that is bigger than, or greater than 3.
So 4, 5, 6 , 7... and so on, forever, are ok.
But 2, 1, 0, -1, -2... and so on are not ok.
And remember, the sign means "greater than"... so in this case y cannot be equal to 3, y must be greater than 3. So, the answer 3 is not ok.
This means y can be any number that is smaller than, or less than 5.
So 4, 3, 2, 1, 0, -1, -2... and so on are ok.
But 6, 7, 8, 9, 10... and so on, forever, are not ok.
The sign means "less than"... so in this case y cannot be equal to 5, y must be less than 5. So, the answer 5 is not ok.
The signs can be used together to say that the value of y is bigger than one number and less than another number... that y lies between two numbers. This means that y must bigger than 3, but less than 7.
So, the numbers that y can be are 4, 5 and 6.
y cannot be 3 or 7, because the signs don't say "equal to".
y cannot 8, 9 10, 11... and so on.
And y cannot be 2, 1, 0 -1, -2... and so on.
This means that y must be less than 5, but bigger than -2.
Note, you can use negative numbers, like -2, in inequalities. You can also use other numbers too, such as fractions or decimals.
So, the numbers that y can be are 4, 3, 2, 1, 0 and -1.
y cannot be either 5 or -2, because the signs don't say "equal to", but "bigger than" and "less than".
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These signs are similar to the ones we've looked at already, except they indicate that the possible range of numbers can include, be equal, to the numbers either side of the inequality. Let's take example 1 above, but change the sign to ³.
This means that y is greater than or equal to 3.
So, y can be 3 and all the numbers bigger than 3... 4, 5, 6, 7, 8 and so on.
Let's take another of the examples above, example 2, but change to sign to £.
This means that y is less than or equal to 5.
So, y can be 5 and all the numbers less than 5... 4, 3, 2, 1, 0, -1...
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All four signs can be used together.
This means that y can be bigger than 3 or equal to 3, and less than 7 or equal to 7.
So, y can be 3, 4, 5, 6 and 7.
This means that y can be bigger than 3 or equal to 3, and less than 7 but not equal to 7.
So, y can be 3, 4, 5, 6 but not 7.
This means that y can be less than 5 or equal to 5, and bigger than -2 or equal to -2.
So, y can be 5, 4, 3, 2, 1, 0, -1 and -2.
This means that y can be less than 5 or equal > to 5, and bigger than -2 but not equal to -2.
So, y can be 5, 4, 3, 2, 1, 0, -1 but not -2.
Bigger than >...
What goes on this side is greater than, bigger than... > what goes on this side is smaller than, less than
Less than <...
What goes on this side is less than, smaller than... < what goes on this side is bigger than, greater than
Some simple examples
Example 1
y > 3
Example 2
y < 5
Using the signs together
The number here is less than..< y, which in turn is smaller than... < the number here is the biggest number
The number here is bigger than..> y, which in turn is bigger than... < the number here is the smallest number
Some examples
Example 3
3 < y < 7
Example 4
5 > y > -2
The bigger-than-or-equal-to and less-than-or-equal-to signs
Bigger than or equal to ³...
What goes on this side is greater than, bigger than... or equal to ³ what goes on this side is smaller than, less than... or equal to
Less than or equal to £...
What goes on this side is less than, smaller than... or equal to £ what goes on this side is bigger than, greater than, or equal to
The bigger-than-or-equal-to sign... ³
Example 5
y ³ 3
The less-than-or-equal-to sign... £
Example 6
y £ 5
Using them together
Some examples
Example 7
3 £ y £ 7
Example 8
3 £ y < 7
Example 9
5 ³ y ³ -2
Example 10
5 ³ y > -2