Inequalities With Variables on Both Sides

1) Goal

When an inequality has the variable on both sides, your job is to collect the variable terms on one side and the numbers on the other.

2) Solve in a repeatable order

1. Expand any parentheses if needed.
2. Move variable terms to one side by adding or subtracting the same expression from both sides.
3. Move constants to the other side.
4. Combine like terms on each side.
5. Isolate the variable by dividing or multiplying.

3) Important inequality rule

If you multiply or divide both sides by a negative number, reverse the inequality sign. This is essential.

4) Example pattern

If you get something like [ ax + b < cx + d, ] first bring the (x)-terms together, then the numbers together, and then solve for (x).

5) Check your answer

Test a value from your solution set in the original inequality. If it makes the statement true, your result is reasonable. Also make sure the final answer is written in a simplified form, such as a single inequality or interval when appropriate.