How to Solve Multi-Step Inequalities? (+FREE Worksheet!)

Algebra 1

How to Solve Multi-Step Inequalities

A multi-step inequality is solved almost exactly like a multi-step equation — distribute, combine like terms, gather the variable — with one rule on top: if you multiply or divide both sides by a negative, flip the inequality sign. We’ll work through it carefully, with a solver, drills, and a worksheet maker a tap away.

Tutor-style math help

Solve Multi-Step Inequalities: what to notice and how to work it

Inequalities skill

Inequalities describe a set of possible values. Solve the boundary like an equation, then decide which side of the boundary makes the statement true.

What to notice first

Watch the comparison sign from the first line to the last. Multiplying or dividing by a negative reverses the direction.

Common student mistake

Do not forget open and closed endpoints. Strict signs use open circles; signs with equals use closed circles.

Key formulas and cues

\(a<b\)

\(a\le b\)

\(\text{multiply/divide by a negative} \Rightarrow \text{reverse the sign}\)

\(|x-a|<b \Rightarrow a-b<x<a+b\)

A reliable path

Solve the boundaryTemporarily treat the inequality like an equation.
Choose the sideUse the sign or test a number if the direction is not obvious.
Graph the solutionUse the correct endpoint and shade the values that work.

Worked examples

Flip the sign

Example: \(-3x>12\)

Divide both sides by -3.
Reverse the inequality sign.
Simplify 12 divided by -3.

Answer: \(x<-4\)

Keep the sign

Example: \(x+5\le9\)

Subtract 5 from both sides.
No negative multiplication or division happened.
Keep the sign direction.

Answer: \(x\le4\)

Open pop-up practice

Try one before moving on

Try: Solve \(-2x\le10\).

Answer: \(x\ge-5\). Divide by -2 and flip the sign.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

⚡ Step-by-Step Solver 📄 Worksheet Maker ✏️ Practice Drills 📇 Flashcards

Illustration of students learning How to Solve Multi-Step Inequalities

If you can solve a multi-step equation, you’re most of the way to solving a multi-step inequality. You do the same moves — distribute, combine like terms, get the variable by itself — but you carry an inequality sign along instead of an equals sign, and there’s one rule that catches almost everyone: multiply or divide by a negative, and the sign flips. The reward is an answer that’s a whole range of values, not a single number.

In short: simplify both sides, gather the variable, then isolate it — and flip the inequality whenever you multiply or divide both sides by a negative. For example, \(3x + 2 > x + 10\) becomes \(x > 4\).

The big idea

Solve Like an Equation, Watch for the Flip

Adding or subtracting never changes the direction of an inequality. Multiplying or dividing by a positive doesn’t either. The only move that reverses the sign is multiplying or dividing both sides by a negative number — because that mirrors every value across zero, reversing their order.

How to solve (4 steps):

Distribute to clear parentheses.
Combine like terms on each side.
Gather the variable on one side; move constants to the other.
Divide to isolate the variable — and flip the sign if you divided by a negative.

Tutor tip: One way to dodge the flip entirely: move the variable to whichever side keeps its coefficient positive. For \(5 – 3x \ge x + 9\), adding \(3x\) to both sides avoids ever dividing by a negative.

Worked Examples

Same moves as a multi-step equation — then read the range off the number line.

Example A — Variables on both sides

Solve \(3x + 2 > x + 10\).

Subtract \(x\) from both sides to gather the variable: \(2x + 2 > 10\).
Subtract 2: \(2x > 8\).
Divide by 2 (positive, no flip): \(x > 4\). Graph: open circle at 4, arrow right.

Answer: \(x > 4\)

Example B — Distribute first

Solve \(2(x – 1) \le 8\).

Distribute the 2: \(2x – 2 \le 8\).
Add 2 to both sides: \(2x \le 10\).
Divide by 2: \(x \le 5\). Graph: closed circle at 5, arrow left.

Answer: \(x \le 5\)

Example C — Divide by a negative (flip!)

Solve \(-2x + 3 < 7\).

Subtract 3 from both sides: \(-2x < 4\).
Divide by \(-2\) — dividing by a negative, so flip: \(x > -2\).
Graph: open circle at \(-2\), arrow right.

Answer: \(x > -2\)

Example D — Negative after gathering

Solve \(5 – 3x \ge x + 9\).

Subtract \(x\) and 5 from both sides: \(-4x \ge 4\).
Divide by \(-4\) and flip: \(x \le -1\).
Graph: closed circle at \(-1\), arrow left.

Answer: \(x \le -1\)

Where You’ll Use It

Multi-step inequalities show up whenever a real situation involves a limit and a few moving parts: staying within a budget after a setup fee, keeping a grade above a cutoff, or making sure a load stays under a weight cap. Solving one tells you the full set of values that keep you on the right side of the line.

Easy Points to Lose

Forgetting to flip. The classic miss — dividing or multiplying by a negative without reversing the sign.
Flipping when you shouldn’t. Adding or subtracting (even a negative) never flips the sign; only multiplying/dividing by a negative does.
Distributing to one term. \(2(x – 1)\) is \(2x – 2\); multiply both terms inside.
Reading the final direction wrong. After a flip, double-check which way the symbol points before you graph it.

Your Turn: Solve

Solve each, flipping when needed, then reveal the answers.

\(4x – 1 < 2x + 7\)
\(3(x + 2) \ge x + 10\)
\(-x + 4 \le 2x – 5\)
\(2x + 9 > 5x\)
\(10 – 2x \le 4\)
\(\dfrac{x}{2} + 1 > 3\)

Show answers (with graphs)

\(\color{blue}{x<4}\)
\(\color{blue}{x\ge 2}\)
\(\color{blue}{x\ge 3 \text{ (flipped)}}\)
\(\color{blue}{x<3 \text{ (flipped)}}\)
\(\color{blue}{x\ge 3 \text{ (flipped)}}\)
\(\color{blue}{x>4}\)

1. \(x < 4\)

2. \(x \ge 2\)

3. \(x \ge 3\)

4. \(x < 3\)

5. \(x \ge 3\)

6. \(x > 4\)

Keep practicing

Make Your Own Inequalities Worksheet

Generate fresh multi-step inequalities with a full answer key — print or save as a PDF.

New problems every click — never the same sheet twice

Step-by-step answer key so you can self-check

📄 Open the Worksheet Maker ⚡ Inequality Solver

⚖️

Frequently Asked Questions

When do I flip the inequality sign?

Only when you multiply or divide both sides by a negative number. Adding or subtracting never flips it.

How is this different from a multi-step equation?

The steps are identical — distribute, combine, isolate — except you keep an inequality sign and flip it when dividing or multiplying by a negative. The answer is a range, not one value.

How can I avoid flipping at all?

Move the variable to the side that keeps its coefficient positive. Then you only ever divide by a positive number, so the sign never flips.

How do I check my answer?

Pick a number inside your solution range and plug it into the original inequality; it should be true. Test one outside the range to confirm it’s false.

Continue Your Study

Ready for the next step? Pick up right where this lesson leaves off:

Up nextLinear InequalitiesContinue with Inequalities and Systems of EquationsOpen lesson →Up nextHow to Solve Systems of EquationsContinue with Inequalities and Systems of EquationsOpen lesson →Up nextSystems of Equations with SubstitutionContinue with Inequalities and Systems of EquationsOpen lesson →

Effortless Math Team

3 weeks ago

Effortless Math Team

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