Solve any multi-step inequality instantly — get step-by-step solutions including distribution, combining like terms, and variable isolation. Free, no sign-up required.
Why this page is built for multi-step intent
Multi-Step Solver
Built for multi-step inequalities that require distribution, combining like terms, or moving variable terms before isolating x — with sign-flip detection and number line output.
Supported Input Styles
3(x+2)>2x-1 distributes first, then moves the variable term to one side.2(x-3)+x<=9 combines like terms after distributing before isolating x.4x-2(x+1)>=6 distributes a negative factor — watch for sign changes inside the parentheses.-2(3x-4)<x+5 distributes a negative coefficient across both terms inside the brackets.5x+3>2(x-1)+4 has variable terms on both sides that must be collected before the final step.Math Keyboard
Tap symbols, numbers, or actions for fast linear-inequality input.
Result
The multi-step solution isolates x > -7.
Step 1
Distribute
Expand any parentheses by multiplying the factor outside into each term inside. Keep track of sign changes when the factor is negative.
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Step 2
Move variable terms to one side
Subtract or add the variable term from one side so all x terms are on the same side.
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Step 3
Move constant terms to the other side
Subtract or add the constant so only the variable term remains on one side.
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Recent History
Saved locally in this browser so you can return to recent multi-step inequalities while checking classwork.
Calculator Types
Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.
Solve two-step inequalities with the sign-flip rule, number line, and interval notation.
Solve multi-step inequalities with distribution, combining like terms, variable collection, and sign-flip detection.
Solve quadratic inequalities with sign analysis, roots, interval notation, and a number-line graph.
Solve absolute value inequalities with case splitting, interval notation, and step-by-step explanations.
Solve compound inequalities with interval intersection, union logic, and graph output.
Solve rational inequalities with steps, excluded values, sign charts, and interval notation.
Zone 4
The fastest way to use this calculator is to type the inequality exactly as it appears on your worksheet. You can enter forms such as 3(x + 2) > 2x - 1, 2(x - 3) + x <= 9, or -2(3x - 4) < x + 5. The parser handles parentheses, negative coefficients, variable terms on both sides, and all four inequality symbols without any special formatting.
After you enter a problem, the Steps tab breaks the work into every individual operation rather than skipping to the final answer. Multi-step inequalities are different from two-step problems because the path to isolating x is longer and the intermediate expressions matter. The steps show distribution first, then combining like terms, then collecting variable terms on one side, and finally dividing out the coefficient. If the last division involves a negative number, the sign-flip rule is called out at that exact step.
If you are checking homework, compare the Steps tab to your own work line by line — multi-step errors usually happen in the distribution or combining step, not the final division. If you are learning the pattern for the first time, work through the steps in order and then open the Graph tab to see what the final interval looks like on a number line.
For shorter first-degree problems, use the two-step inequality calculator or the linear inequality calculator.
Enter a multi-step inequality such as 3(x + 2) > 2x - 1 or -2(3x - 4) < x + 5.
Review each step card to see distribution, combining like terms, variable collection, and the final isolation — including any sign flip.
Open the graph, interval notation, and verify tabs to confirm the answer from multiple angles.
A multi-step inequality is a one-variable inequality that requires more than two operations to isolate the variable. The extra steps typically come from one or more of three sources: parentheses that need to be distributed, like terms on the same side that need to be combined, or variable terms on both sides that need to be collected onto one side before the final isolation step.
A typical example is 3(x + 2) > 2x - 1. Solving it requires distributing the 3, then subtracting 2x from both sides to collect the variable terms, and then subtracting 6 to isolate x. None of those three moves is the final step on its own — they all have to happen in the right order before the answer appears.
Multi-step inequalities follow the same sign-flip rule as simpler inequalities: if you divide or multiply both sides by a negative number at any point in the process, the inequality symbol reverses. The rule applies at that specific step regardless of how many steps came before it. That is why the calculator labels the sign-flip step explicitly rather than only showing the final result.
| Comparison point | Two-step inequality | Multi-step inequality |
|---|---|---|
| Operations needed | Exactly 2 | 3 or more |
| Parentheses | Not required | Often present — must distribute first |
| Variable terms | One side only | May appear on both sides |
| Like terms | Not required | May need combining before isolating |
| Sign flip risk | One place (final division) | Can occur at any division step |
Multi-step inequalities follow a consistent order of operations. Working through that order reliably — distribute, combine, collect, isolate — prevents the most common errors. The sign-flip rule applies at the isolation step just as it does in simpler inequalities. For a broader rule overview, see how to solve inequalities.
If any term has parentheses, expand them before doing anything else. Multiply the factor outside the parentheses into every term inside. Pay close attention when the factor is negative — every sign inside the parentheses flips.
Example
After distributing, look for like terms on the same side of the inequality and combine them. Do the left side first, then the right side. Do not move terms across the inequality symbol during this step.
Example
If variable terms appear on both sides, add or subtract to move them all to one side. Choose the side that keeps the coefficient positive when you can.
Example
Move the remaining constant to the other side, then divide by the coefficient of x. If the coefficient is negative, flip the inequality symbol. This is the same final step as in a two-step inequality.
Example
When the factor outside parentheses is negative, every sign inside changes. One missed sign usually breaks every later step.
Only combine like terms on the same side first. Terms from opposite sides should be moved with addition or subtraction, not merged directly.
In long problems, the negative coefficient often appears only near the end. Pause before the last division and check whether the inequality must reverse.
These three walkthroughs show the main branches of multi-step work: straight distribution, combining like terms, and a hidden sign-flip case after the algebra simplifies.
This is the cleanest multi-step pattern: distribute first, move the variable term, then move the constant.
Distribute the 3
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Subtract 2x from both sides
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Subtract 6 from both sides
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Answer
Interval
Set notation
This version adds a combining step after distribution before the inequality can be isolated.
Distribute the 2
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Combine the x-terms
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Isolate x
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Answer
Interval
Set notation
This example shows why multi-step work hides the sign-flip moment: it appears only after distribution and moving terms.
Distribute the -2
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Move x and constants
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Divide by -7 and flip the sign
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Answer
Interval
Set notation
A multi-step inequality still ends as a single ray once x is isolated. The extra work happens in the algebra, not in the graph. After 3(x + 2) > 2x - 1 simplifies to x > -7, the graph is the same open-end ray you would draw for any strict greater-than statement.
The endpoint style still follows the final symbol: open for < or >, closed for <= or >=. That visual rule is worth checking after long algebra because a correct graph often catches a sign-flip mistake immediately.
Strict inequality
Inclusive inequality
Interval notation is the compact way to record the final ray after the multi-step algebra is done. The notation rules do not change just because the solving process was longer: parentheses exclude a boundary, brackets include it, and infinity always uses a parenthesis.
Once you isolate x, translate the last inequality directly. That makes interval notation a good final check: if your graph and interval do not agree, one of the earlier algebra steps probably drifted.
The full interval notation guide explains the bracket and parenthesis rules in more detail.
| Inequality | Interval notation | Number line | Meaning |
|---|---|---|---|
| Values greater than a, but not including a. | |||
| Values greater than a, including a. | |||
| Values less than a, but not including a. | |||
| Values less than a, including a. |
It requires more than two operations to isolate the variable — typically because of parentheses, like terms, or variable terms on both sides.
Yes. The rule applies at the exact step where you divide or multiply by a negative number, regardless of how many steps came before it.
Distribute parentheses first, then combine like terms on each side, then move variable terms to one side, then move constants to the other side, then divide by the coefficient.
Yes, and that is one of the defining features of many multi-step inequalities. Subtract the smaller variable term from both sides to keep the coefficient positive.
The two-step calculator handles exactly two operations. This calculator handles the full sequence including distribution and like-term combining that must happen before the two-step pattern begins.
Work from the innermost parentheses outward, distributing one layer at a time. The calculator handles nested parentheses automatically.
Zone 6
Keep the same algebra rule set nearby while you practice distribution, sign flips, and final interval checks.