3x + 5 < 20
x is multiplied by 3, then 5 is added.
Free Algebra Guide
Two-Step Inequalities
Learn how to solve two-step inequalities by undoing the constant first, then the coefficient or divisor, while knowing exactly when the inequality symbol must reverse.
A two-step inequality requires two inverse operations to isolate the variable. In most problems, remove the constant first and then undo the coefficient or divisor. The special inequality rule still applies: if you multiply or divide both sides by a negative number, reverse the inequality symbol.
On This Page
- What is a two-step inequality?
- Why the solving order matters
- Step 1: remove the constant
- Step 2: remove the coefficient or divisor
- When to reverse the inequality sign
- Two-step inequalities with division
- Fractional solutions
- Graphing the solution
- Checking a solution
- Worked examples
- Common mistakes
- A reliable strategy
- Interactive practice
What Is a Two-Step Inequality?
A two-step inequality usually has both a constant term and a coefficient or divisor attached to the variable.
3x + 5 < 20
First remove the constant.
Subtract 5 from both sides.
Subtract 5 from both sides.
Then remove the coefficient.
Divide both sides by 3.
Divide both sides by 3.
The inequality symbol stays the same here.
Both operations use addition/subtraction or a positive divisor.
Both operations use addition/subtraction or a positive divisor.
3x + 5 < 20 → 3x < 15 → x < 5
Why the Solving Order Matters
To isolate the variable, undo the operations in reverse order from how they are applied to the variable.
Remove +5, then ×3
Undo the outer operation first, then the operation attached directly to x.
Think Outside In
The constant is usually farther from the variable than the coefficient, so remove the constant first.
Remove the Constant Term First
Use addition or subtraction on both sides to remove the constant. This step does not reverse the inequality symbol.
Example: 4x − 7 ≥ 13
Start
4x − 7 ≥ 13
Add 7
4x ≥ 20
Do Not Flip the Symbol Yet
Adding or subtracting never requires an inequality sign reversal.
Remove the Coefficient or Divisor
Once the constant is gone, undo the multiplication or division attached directly to the variable.
4x ≥ 20
Divide both sides by 4 → x ≥ 5
x/3 < 6
Multiply both sides by 3 → x < 18
Check the Sign of the Number
If the multiplier or divisor is positive, keep the inequality symbol. If it is negative, reverse the symbol.
When Do You Reverse the Inequality Sign?
Reverse the inequality symbol only when you multiply or divide both sides by a negative number.
Example: −2x + 5 < 13
Remove +5
−2x < 8
Divide by −2
x > −4
The Flip Happens on the Negative Step
The first step kept < because subtraction does not reverse the symbol. The second step changed < to > because the inequality was divided by −2.
−2x + 5 < 13 → −2x < 8 → x > −4
Two-Step Inequalities With Division
If the variable is divided by a number, remove the constant first and then multiply both sides by the divisor.
x/4 + 3 > 8
x/4 > 5 → x > 20
−x/3 + 2 < 6
−x/3 < 4 → x > −12
Two-Step Inequalities With Fractional Solutions
If division does not produce a whole number, keep the result as a reduced fraction.
Example: 4x + 3 > 13
Subtract 3
4x > 10
Divide by 4
x >
5
2
Graphing Two-Step Inequality Solutions
Once the variable is isolated, graph the final inequality on a number line just as you would for a one-step inequality.
| Final Solution | Endpoint | Shade |
|---|---|---|
| x < 5 | Open circle at 5 | Left |
| x > −4 | Open circle at −4 | Right |
| x ≤ 3 | Closed circle at 3 | Left |
| x ≥ −2 | Closed circle at −2 | Right |
How to Check a Two-Step Inequality Solution
Choose a number from the final solution set and substitute it into the original inequality.
Check x < 5 for 3x + 5 < 20
Choose x = 4 because 4 is less than 5.
3(4) + 5 < 20
17 < 20 is true, so x = 4 belongs to the solution set.
Worked Two-Step Inequality Examples
3x + 5 < 20
- Subtract 5: 3x < 15.
- Divide by 3: x < 5.
5x − 3 > 17
- Add 3: 5x > 20.
- Divide by 5: x > 4.
−2x + 5 < 13
- Subtract 5: −2x < 8.
- Divide by −2 and reverse: x > −4.
−3x + 2 > −10
- Subtract 2: −3x > −12.
- Divide by −3 and reverse: x < 4.
x/4 + 3 > 8
- Subtract 3: x/4 > 5.
- Multiply by 4: x > 20.
4x + 3 > 13
- Subtract 3: 4x > 10.
- Divide by 4: x > 5/2.
Common Mistakes With Two-Step Inequalities
Undoing the Coefficient First
In a standard two-step inequality, remove the constant before the coefficient.
Flipping Too Early
Do not reverse the inequality during addition or subtraction. Reverse only when multiplying or dividing by a negative.
Forgetting the Final Flip
If the final coefficient or divisor is negative, the inequality symbol must reverse.
Changing Only One Side
Use the same inverse operation on both sides to keep the inequality balanced.
Skipping the Intermediate Inequality
Write the updated inequality after removing the constant before moving to the coefficient step.
Steps for Solving Two-Step Inequalities
1. Identify the constant
Find the addition or subtraction term outside the variable term.
2. Remove the constant
Use the inverse addition or subtraction on both sides.
3. Identify the coefficient or divisor
Look at the multiplication or division attached directly to the variable.
4. Undo the coefficient or divisor
Use division or multiplication on both sides.
5. Check for a negative operation
If you multiplied or divided by a negative, reverse the inequality symbol.
6. Graph or check if needed
Show the solution set on a number line or test a value in the original inequality.
Remove the constant first. Then undo the coefficient. Reverse only for negative multiplication or division.
Practice Two-Step Inequalities
Use the interactive inequalities tool and choose Two-Step to practice constants, coefficients, negative values, division forms, fractions, and sign reversals.
Open the Solving Inequalities ToolRelated Math Resources
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