How to write down all the integers that satisfy an inequality between two numbers.

If you are asked to write down all the integers that satisfy the inequality -3 < x ≤ 4, then you are looking for all the values of x that are more -3 and less than or equal to 4.

This is because -3 < x means x > -3 (x is more than 3) and x ≤ 4 means x is less than or equal to 4.

Also an integer is a whole number so you don’t need to write down any decimals or fractions.

So the integers that satisfy -3 < x ≤ 4 are -2,-1, 0, 1, 2, 3 and 4.

Let’s take a look at another question:

Write down all the integers that satisfy the inequality -2 ≤ x < 3.

Here, -2 ≤ x means x ≥ -2 so you want all the integers more than or equal to -2.

Also x < 3 means all the integers less than 3.

So the required integers are -2,-1, 0, 1 and 2.


Let’s take a look at a harder example.

Write down all the integers that satisfy -6 ≤ 2x ≤ 5

This time you have 2x in the centre of the inequality, so the first thing to do is divide everything by 2.

This gives:

-3 ≤ x ≤ 2.5

Now -3 ≤ x is the same as x ≥ -3 so you need all the integers more than or equal to -3.

Also x ≤ 2.5 means you need all the integers less than or equal to 2.5 (don’t include 2.5 as 2.5 is a decimal and not an integer).

So the required integers are:

-3, -2, -1, 0, 1 and 2.

For some other maths help take a look at some of my other hubs:

Adding (or taking) two fractions together. Step by step fraction guide.

Working out percentages of amounts using a calculator method.

Stationary points. How to find a stationary point on a graph (turning points)

How to complete the square on a quadratic equation (completing the square).

Converting between millimetres and metres (metric units of length 1000mm = 1m)

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