What Is a Linear Inequality?
An inequality is like an equation, but instead of saying two things are equal, it says one is greater than, less than, greater than or equal to, or less than or equal to another.
A mathematical statement that compares two linear expressions using one of the symbols , , , or . For example, describes a speed limit of 25 km/h — any speed at or below 25 is acceptable.
The four inequality symbols are:
- — strictly less than
- — strictly greater than
- — less than or equal to
- — greater than or equal to
Think of a speed limit sign on a road near roadworks. If the limit is 25 km/h, you can travel at 10, 20, or exactly 25 — but not 26. This is exactly what means: can be any value up to and including 25.
Notice that and say the same thing — just from opposite sides. This matters when you rearrange inequalities!
Rules for Handling Inequalities
Solving inequalities works almost exactly like solving equations — with two important exceptions.
Rules that keep the inequality sign the same:
- Adding or subtracting the same number to both sides → sign stays the same
- e.g. if , then ✓
- Multiplying or dividing both sides by a positive number → sign stays the same
- e.g. if , then ✓
Rules that REVERSE the inequality sign:
- Interchanging the left-hand side (LHS) and right-hand side (RHS) → sign reverses
- e.g. becomes
- Multiplying or dividing both sides by a negative number → sign reverses
- e.g. if , then , i.e. ✓
The most common mistake students make is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. This is the one rule that has no parallel in solving regular equations — always watch out for it!
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