Integer solutions to inequalities

When solving inequalities there will be a range of answers. Because any numbers represented by the range are acceptable, there are an infinite amount of solutions to inequalities.

For example, if a \textgreater 3, then any number that is bigger than 3 is a possible answer, from any decimal slightly bigger than 3 to infinity.

Sometimes, only integer solutions are required.

Example

List the integer solutions to -5 \textless k \textless 1.

This can be read as -5 is less than k, which is less than 1.

That means that k is larger than -5, but not equal to -5, so the smallest integer that k can be is -4.

k is less than 1, but not equal to 1, so the largest integer that k can be is 0.

k can also be the integers between -4 and 0.

This means that the integer solutions to -5 \textless k \textless 1 are: -4, -3, -2, -1, 0.

Question

List the integer solutions of -3 \textless 2 e \leq 8.

The variable e in the centre of the inequalities has been multiplied by 2 to give 2e. To find the solutions for e, use inverse operations and divide all terms by 2.

\begin{array}{ccccc} -3 & \textless & 2e & \leq & 8 \\ \div 2 && \div 2 && \div 2 \\ -1.5 & \textless & e & \leq & 4 \end{array}

e is larger than -1.5, so the smallest integer solution is -1. e is less than or equal to 4, so the largest integer solution is 4. e can also be any number between these two solutions.

The integer solutions of -3 \textless 2 e \leq 8 are: -1, 0, 1, 2, 3, 4.