Multi-Step Inequalities

Q: When exactly do I flip the sign?

Only when you multiply or divide both sides of the inequality by a negative number — which, if it happens at all, is the last step. Distributing, combining like terms, and adding or subtracting never flip the sign.

Q: Does distributing a negative flip the sign?

No. Distributing $-3$ across $(x - 2)$ changes that one side to $-3x + 6$, but you haven't done anything to both sides of the inequality, so the symbol stays. The flip only applies to operations performed on both sides.

Q: What order should the steps go in?

Distribute, combine like terms on each side, move the constant with adding or subtracting, then divide by the coefficient. It's the same order as multi-step equations — inequalities just add the flip check at the final divide.

Q: How do I check an answer like $x \geq 4$?

Test two numbers in the original inequality: one that should work (like $5$) and one that shouldn't (like $0$). If the first comes out true and the second false, your answer and its direction are both right.

A multi-step inequality is a two-step inequality with some cleanup in front. Problems like $2(x + 3) > 16$ or $-4x + 8 - 2x \leq -16$ need you to distribute or combine like terms first — after that, they solve exactly like the shorter ones.

The plan is always the same: simplify first, then solve. And the sign-flip rule still applies at exactly one moment — flip the inequality symbol only when you multiply or divide both sides by a negative number, never during the simplifying steps.

Simplify first, then solve

Step 1: clean up each side on its own. Distribute across any parentheses, then combine like terms. Nothing crosses the inequality symbol during this step, so the symbol can't flip here.

Step 2: solve the simplified inequality like a two-step problem — move the constant with adding or subtracting, then divide by the coefficient.

Step 3: at that final divide (or multiply), check the sign of the number you're dividing by. Positive: the symbol stays. Negative: flip it.

Where the flip does and doesn't happen

Combining like terms never flips the sign, even when the terms are negative. In $-4x + 8 - 2x \leq -16$ , combining $-4x$ and $-2x$ into $-6x$ is just rewriting the left side — the symbol doesn't move.

The flip only happens when both sides get multiplied or divided by a negative. In that same problem you eventually reach $-6x \leq -24$ , and dividing both sides by $-6$ is the moment the $\leq$ becomes $\geq$ : the answer is $x \geq 4$ .

So there is at most one flip per problem, and it always comes at the very end. If you flipped anywhere earlier, that's the bug.

Worked examples

Example 1: distribute first

Solve $2(x + 3) > 16$ .

Start with the inequality

2(x + 3) > 16

Distribute the

2

2x + 6 > 16

Subtract

6

from both sides

2x > 10

Divide both sides by

2

— positive, no flip

x > 5

Answer: $x > 5$

Example 2: combine like terms, then flip

Solve $-4x + 8 - 2x \leq -16$ .

Start with the inequality

-4x + 8 - 2x \leq -16

Combine the

x

terms

-6x + 8 \leq -16

Subtract

8

from both sides

-6x \leq -24

Divide both sides by

-6

and flip the sign

x \geq 4

Answer: $x \geq 4$

Example 3: distribute and a constant outside

Solve $5(3x - 4) - 7 < 33$ .

Start with the inequality

5(3x - 4) - 7 < 33

Distribute the

5

15x - 20 - 7 < 33

Combine the constants

15x - 27 < 33

Add

27

to both sides

15x < 60

Divide both sides by

15

— positive, no flip

x < 4

Answer: $x < 4$

Try one yourself

Common questions

When exactly do I flip the sign?

Only when you multiply or divide both sides of the inequality by a negative number — which, if it happens at all, is the last step. Distributing, combining like terms, and adding or subtracting never flip the sign.

Does distributing a negative flip the sign?

No. Distributing $-3$ across $(x - 2)$ changes that one side to $-3x + 6$ , but you haven't done anything to both sides of the inequality, so the symbol stays. The flip only applies to operations performed on both sides.

What order should the steps go in?

Distribute, combine like terms on each side, move the constant with adding or subtracting, then divide by the coefficient. It's the same order as multi-step equations — inequalities just add the flip check at the final divide.

How do I check an answer like

x \geq 4

Test two numbers in the original inequality: one that should work (like $5$ ) and one that shouldn't (like $0$ ). If the first comes out true and the second false, your answer and its direction are both right.

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Multi-Step Inequalities

Simplify first, then solve

Where the flip does and doesn't happen

Worked examples

Example 1: distribute first

Example 2: combine like terms, then flip

Example 3: distribute and a constant outside

Try one yourself

Common questions

Keep learning

One-Step Inequalities

Inequalities with Variables on Both Sides

Introduction to Inequalities

Equations with Variables on Both Sides