Linear Inequality Solver – Solve and Graph Inequalities Online
Solve linear inequalities instantly with our free online inequality solver. Get solutions displayed on a number line with clear step-by-step explanations for all inequality types.
Inequality Format:
ax + b operator c
Understanding Linear Inequalities
A linear inequality is like a linear equation, but instead of an equals sign, it uses inequality symbols: <, >, ≤, or ≥. The solution to an inequality isn't a single number – it's a range of values that make the inequality true.
Solving inequalities is similar to solving equations: isolate the variable. But there's one crucial difference – when you multiply or divide by a negative number, you must flip the inequality sign. This is the most common mistake students make.
How to Solve Linear Inequalities
- 1
Simplify both sides
Combine like terms and distribute if needed.
- 2
Move variable terms to one side
Add or subtract to get all x terms on one side, constants on the other.
- 3
Isolate the variable
Divide by the coefficient. If it's negative, FLIP the inequality sign!
- 4
Express the solution
Write as an inequality, in interval notation, and graph on a number line.
Worked Examples
Example 1: Simple Inequality
Example 2: Negative Coefficient (Flip!)
Example 3: Variables on Both Sides
Example 4: Special Cases
No solution: 2x + 5 < 2x + 1
No value of x makes this true.
All real numbers: 3(x+2) ≥ 3x + 6
Every value of x works!
Quick Fact
The inequality symbols < and > were introduced by Welsh mathematician Robert Recorde in 1557. He also invented the equals sign (=), saying "no two things can be more equal" than parallel lines. The ≤ and ≥ symbols came later, in the 18th century.
Frequently Asked Questions
Why do I flip the sign when dividing by negative?
Think about it: 3 < 5 is true. Multiply both by -1: -3 and -5. Now -3 is GREATER than -5, so -3 > -5. The relationship reverses because negatives flip the order on the number line.
What's the difference between < and ≤?
< means "strictly less than" – the boundary is NOT included. ≤ means "less than or equal to" – the boundary IS included. On a number line, use open circle for < and closed circle for ≤.
How do I check my solution?
Pick a value in your solution range and plug it into the original inequality. It should make the inequality true. Also test a value outside the range – it should make the inequality false.
What is interval notation?
A compact way to write solution sets. Parentheses ( ) mean the endpoint is excluded. Brackets [ ] mean it's included. ∞ always uses parentheses since infinity isn't a specific number.
Can inequalities have no solution?
Yes. If simplifying leads to a false statement like 5 < 1, there's no solution. The solution set is empty (∅). This happens when the inequality is a contradiction.
What about compound inequalities?
Compound inequalities like 2 < x < 5 mean x is between 2 and 5. Solve by applying operations to all three parts. For "or" inequalities, solve each part separately and combine the solutions.
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