x + 6 < 14
x < 8
Subtract 6 from both sides.
Free Algebra Guide
One-Step Inequalities
Learn how to solve one-step inequalities with inverse operations, understand what the inequality symbols mean, and know exactly when the inequality sign must reverse.
A one-step inequality can be solved with one inverse operation. The process is very similar to solving a one-step equation, but inequalities have one critical extra rule: when you multiply or divide both sides by a negative number, the inequality symbol reverses.
On This Page
- What is a one-step inequality?
- Understanding inequality symbols
- Addition and subtraction inequalities
- Multiplication and division inequalities
- When to reverse the inequality sign
- Fractions and decimal solutions
- Graphing inequality solutions
- Checking a solution
- Worked examples
- Common mistakes
- A reliable strategy
- Interactive practice
What Is a One-Step Inequality?
A one-step inequality is an inequality in which the variable can be isolated using one inverse operation.
x + 5 < 12
The variable has 5 added to it.
Undo that addition with subtraction.
Undo that addition with subtraction.
Subtract 5 from both sides.
x < 7
x < 7
The solution is a range of numbers.
Every number less than 7 is a solution.
Every number less than 7 is a solution.
x + 5 < 12 → x < 7
Understanding Inequality Symbols
| Symbol | Meaning | Example |
|---|---|---|
| < | Less than | x < 4 |
| > | Greater than | x > 4 |
| ≤ | Less than or equal to | x ≤ 4 |
| ≥ | Greater than or equal to | x ≥ 4 |
Inequalities Usually Have Many Solutions
An equation such as x = 4 has one solution. An inequality such as x < 4 represents every number less than 4.
One-Step Inequalities With Addition and Subtraction
Addition and subtraction work exactly like they do in equations. Use the inverse operation on both sides. The inequality symbol does not reverse.
x − 5 ≥ 3
x ≥ 8
Add 5 to both sides.
One-Step Inequalities With Multiplication and Division
When the variable is multiplied or divided by a positive number, use the inverse operation and keep the inequality symbol pointing the same direction.
3x < 18
Divide both sides by 3 → x < 6
x/4 ≥ 5
Multiply both sides by 4 → x ≥ 20
Positive Numbers Do Not Reverse the Symbol
Multiplying or dividing both sides by a positive number preserves the direction of the inequality.
When Do You Reverse the Inequality Sign?
Reverse the inequality symbol whenever you multiply or divide both sides by a negative number.
Example: −2x < 8
Start
−2x < 8
Divide by −2
x > −4
Why Does the Symbol Reverse?
Multiplying by a negative reverses the order of numbers on the number line.
For example, 2 < 4 is true. Multiplying both sides by −1 gives −2 > −4. The symbol must reverse to keep the statement true.
Multiply or divide by a negative → reverse < and >, or reverse ≤ and ≥.
One-Step Inequalities With Fractions
Some one-step inequalities produce fractional solutions. Keep the fraction exact and reduce it when possible.
Example: 4x > 10
Divide by 4
x >
10
4
Reduce
x >
5
2
Graphing One-Step Inequality Solutions
Inequality solutions are often shown on a number line. The endpoint tells you where the solution begins, and the arrow shows which direction the solutions continue.
| Solution | Endpoint | Shade |
|---|---|---|
| x < 4 | Open circle at 4 | Left |
| x > 4 | Open circle at 4 | Right |
| x ≤ 4 | Closed circle at 4 | Left |
| x ≥ 4 | Closed circle at 4 | Right |
Open vs. Closed Circles
Use an open circle for < or >. Use a closed circle for ≤ or ≥ because the endpoint itself is included.
How to Check an Inequality Solution
Because an inequality usually has many solutions, choose a number from the solution set and substitute it into the original inequality.
Check x < 7 for x + 5 < 12
Choose a value less than 7, such as x = 6.
6 + 5 < 12
11 < 12 is true, so x = 6 is part of the solution set.
Worked One-Step Inequality Examples
x + 8 < 15
- Subtract 8 from both sides.
- x < 7.
x − 4 ≥ 9
- Add 4 to both sides.
- x ≥ 13.
5x ≤ 20
- Divide both sides by 5.
- x ≤ 4.
−3x > 15
- Divide both sides by −3.
- Reverse > to <.
- x < −5.
x/4 > 3
- Multiply both sides by 4.
- x > 12.
4x > 10
- Divide both sides by 4.
- x > 10/4.
- Reduce: x > 5/2.
Common Mistakes With One-Step Inequalities
Forgetting to Reverse the Symbol
If you multiply or divide by a negative number, the inequality direction must reverse.
Reversing the Symbol for Addition
Adding or subtracting a negative does not automatically reverse the inequality.
Using the Wrong Inverse Operation
Undo addition with subtraction, subtraction with addition, multiplication with division, and division with multiplication.
Treating the Solution Like One Number
x < 5 represents every number less than 5, not only one value.
Using the Wrong Number-Line Endpoint
Use an open circle for strict inequalities and a closed circle when equality is included.
Steps for Solving One-Step Inequalities
1. Identify the operation
Determine what operation is attached to the variable.
2. Use the inverse operation
Apply the inverse operation to both sides.
3. Check for a negative multiplier or divisor
If you multiplied or divided by a negative number, reverse the inequality symbol.
4. Write the solution
Make sure the variable is isolated.
5. Graph if needed
Use the correct endpoint and shade in the correct direction.
6. Check a value
Substitute a value from the solution set into the original inequality.
Add or subtract → keep the symbol. Multiply or divide by a negative → reverse the symbol.
Practice One-Step Inequalities
Use the interactive inequalities tool and choose One-Step to practice inverse operations, negative values, fractions, and sign reversals.
Open the Solving Inequalities ToolRelated Math Resources
Personalized Math Support
Need Help Making Sense of the Math?
If a concept still feels confusing, you do not have to work through it alone. Tell me what you are studying and where you are getting stuck, and I will help you determine the best next step.