Expression Work
Distribute and combine like terms before using inverse operations.
Free Algebra Guide
Multi-Step Inequalities
Learn how to solve multi-step inequalities using distribution, combining like terms, inverse operations, variables on both sides, fractions, and the special rule for multiplying or dividing by negative numbers.
Multi-step inequalities combine several algebra skills in one problem. Before isolating the variable, you may need to distribute, combine like terms, simplify both sides, or move variable terms. After simplifying, solve with inverse operations. Remember that the inequality symbol reverses only when you multiply or divide both sides by a negative number.
On This Page
- What makes an inequality multi-step
- Why you simplify before solving
- Using the distributive property
- Combining like terms
- Variables on both sides
- Using inverse operations
- When to reverse the inequality
- Fractional coefficients and solutions
- Graphing solutions
- Checking solutions
- Worked examples
- Common mistakes
- A reliable strategy
- Interactive practice
What Is a Multi-Step Inequality?
A multi-step inequality requires several algebra steps before the variable is isolated. The problem may contain parentheses, multiple variable terms, constants that can be combined, or variables on both sides.
3(x + 4) + 2x < 27
Parentheses appear first.
Use the distributive property.
Use the distributive property.
Like terms remain afterward.
Combine the x-terms.
Combine the x-terms.
Then solve the simplified inequality.
Undo the constant and coefficient.
Undo the constant and coefficient.
3(x + 4) + 2x < 27 → 3x + 12 + 2x < 27 → 5x + 12 < 27 → 5x < 15 → x < 3
Why Simplification Comes First
Trying to isolate the variable before simplifying often creates unnecessary work. First simplify each side of the inequality as much as possible.
Inequality Work
Once simplified, use inverse operations to isolate the variable.
Simplifying Does Not Reverse the Inequality
Distributing or combining like terms does not automatically change the inequality direction. The symbol reverses only when both sides are multiplied or divided by a negative number.
Using the Distributive Property
Multiply the factor outside the parentheses by every term inside. Keep the inequality symbol unchanged during the simplification.
Example: 3(x + 4) + 2x < 27
Start
3(x + 4) + 2x < 27
Distribute 3
3x + 12 + 2x < 27
Multiply Every Term Inside
3(x + 4) becomes 3x + 12, not 3x + 4.
Combining Like Terms
After distribution, combine terms with identical variable parts. Constants can also be combined with other constants.
Current inequality
3x + 12 + 2x < 27
Combine x-terms
5x + 12 < 27
Keep Unlike Terms Separate
5x and 12 cannot be combined because one is a variable term and the other is a constant.
Inequalities With Variables on Both Sides
If variable terms appear on both sides, use addition or subtraction to move them to one side before finishing the solution.
Example: 5x + 4 < 2x + 19
Subtract 2x
3x + 4 < 19
Subtract 4
3x < 15
Divide by 3
x < 5
Addition and Subtraction Do Not Flip the Sign
Moving a variable term by adding or subtracting the same term on both sides does not reverse the inequality.
Using Inverse Operations
Once the inequality is simplified, remove the constant and then undo the coefficient or divisor.
| Current Inequality | Next Step | Result |
|---|---|---|
| 5x + 12 < 27 | Subtract 12 | 5x < 15 |
| 5x < 15 | Divide by 5 | x < 3 |
| −3x < 12 | Divide by −3 | x > −4 |
| x/4 ≥ 6 | Multiply by 4 | x ≥ 24 |
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 − 5x + 4 < 16
Combine x-terms
−3x + 4 < 16
Subtract 4
−3x < 12
Divide by −3
x > −4
Notice When the Flip Happens
Combining like terms did not reverse the symbol. Subtracting 4 did not reverse the symbol. Only the final division by −3 changed < to >.
Multiply or divide by a negative → reverse < and >, or reverse ≤ and ≥.
Multi-Step Inequalities With Fractions
Fractional coefficients follow the same process. Simplify the fractional terms first, then use inverse operations.
Example: 1 2 x + 3 2 x + 4 < 16
Combine like terms
2x + 4 < 16
Subtract 4
2x < 12
Divide by 2
x < 6
Negative Fraction Example
−1/2x < 4 → x > −8
Multiplying both sides by −2 reverses the inequality.
Graphing Multi-Step Inequality Solutions
After completing all algebra steps, graph the final inequality on a number line.
| Final Solution | Endpoint | Shade |
|---|---|---|
| x < 3 | Open circle at 3 | Left |
| x > −4 | Open circle at −4 | Right |
| x ≤ 5 | Closed circle at 5 | Left |
| x ≥ −2 | Closed circle at −2 | Right |
Open vs. Closed Circles
Use an open circle for < or >. Use a closed circle for ≤ or ≥ because the endpoint is included.
How to Check a Multi-Step Inequality Solution
Choose a value from the solution set and substitute it into the original inequality.
Check x < 3 in 3(x + 4) + 2x < 27
Choose x = 2 because 2 is less than 3.
3(2 + 4) + 2(2) < 27
18 + 4 < 27, so 22 < 27 is true.
Worked Multi-Step Inequality Examples
3x + 2x + 4 < 24
- Combine: 5x + 4 < 24.
- Subtract 4: 5x < 20.
- Divide by 5: x < 4.
4(x − 2) + x ≤ 17
- Distribute: 4x − 8 + x ≤ 17.
- Combine: 5x − 8 ≤ 17.
- Add 8, then divide by 5: x ≤ 5.
2x − 5x + 4 < 16
- Combine: −3x + 4 < 16.
- Subtract 4: −3x < 12.
- Divide by −3 and reverse: x > −4.
−4(x − 2) + 2x ≤ 18
- Distribute: −4x + 8 + 2x ≤ 18.
- Combine: −2x + 8 ≤ 18.
- Subtract 8: −2x ≤ 10.
- Divide by −2 and reverse: x ≥ −5.
5x + 4 < 2x + 19
- Subtract 2x: 3x + 4 < 19.
- Subtract 4: 3x < 15.
- Divide by 3: x < 5.
−3/4x + 1/4x + 2 < 6
- Combine: −1/2x + 2 < 6.
- Subtract 2: −1/2x < 4.
- Multiply by −2 and reverse: x > −8.
1/2(4x + 6) + x > 12
- Distribute: 2x + 3 + x > 12.
- Combine: 3x + 3 > 12.
- Subtract 3: 3x > 9.
- Divide by 3: x > 3.
2(x + 3) + 3(x + 4) < 28
- Distribute: 2x + 6 + 3x + 12 < 28.
- Combine: 5x + 18 < 28.
- Subtract 18: 5x < 10.
- Divide by 5: x < 2.
Common Mistakes With Multi-Step Inequalities
Solving Before Simplifying
Distribute and combine like terms before trying to isolate the variable.
Distributing to Only One Term
The outside factor must multiply every term inside the parentheses.
Combining Unlike Terms
Variable terms and constants cannot be combined unless their variable parts match.
Flipping the Sign During Addition or Subtraction
Adding or subtracting the same value on both sides does not reverse the inequality.
Forgetting the Negative Sign Flip
If you multiply or divide both sides by a negative number, the inequality symbol must reverse.
Losing a Negative During Distribution
Carefully multiply every term when the factor outside the parentheses is negative.
Skipping Algebra Steps
Write one complete updated inequality at a time so errors are easier to identify.
Steps for Solving Multi-Step Inequalities
1. Distribute if needed
Remove parentheses by multiplying the outside factor by every term inside.
2. Combine like terms
Simplify matching variable terms and constants on each side.
3. Move variable terms if needed
If variables appear on both sides, use addition or subtraction to gather them on one side.
4. Remove the constant
Use addition or subtraction on both sides.
5. Remove the coefficient or divisor
Use multiplication, division, or a reciprocal to isolate the variable.
6. Check whether the symbol must reverse
Reverse the inequality only if you multiplied or divided by a negative number.
7. Graph the solution
Use an open or closed endpoint and shade in the correct direction.
8. Check a value
Substitute a value from the solution set into the original inequality.
Simplify first. Solve second. Reverse only for negative multiplication or division.
Practice Solving Multi-Step Inequalities
Open the interactive inequalities tool and choose Multi-Step to practice distribution, like terms, negative coefficients, fractions, and inequality sign reversals one algebra step at a time.
Open the Solving Inequalities ToolRelated Math Resources
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