How to Solve Multi-Step Inequalities? (+FREE Worksheet!)
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.
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 \(x > 4\).
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.
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)
- \(\color{blue}{x<4}\)
- \(\color{blue}{x\ge 2}\)
- \(\color{blue}{x\ge 3 \text{ (flipped)}}\)
- \(\color{blue}{x<3 \text{ (flipped)}}\)
- \(\color{blue}{x\ge 3 \text{ (flipped)}}\)
- \(\color{blue}{x>4}\)
Make Your Own Inequalities Worksheet
Generate fresh multi-step inequalities with a full answer key — print or save as a PDF.
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.
Related Topics
Continue Your Study
Ready for the next step? Pick up right where this lesson leaves off:
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