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

Algebra 1

How to Solve One-Step Inequalities

A one-step inequality is solved almost exactly like a one-step equation — with one famous twist: multiply or divide by a negative and you flip the inequality sign. Learn that single rule and the rest is familiar. Solver, drills, and a worksheet maker are a tap away.

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

Illustration of students learning How to Solve One-Step Inequalities

A one-step inequality is an inequality you solve in a single step — undo one operation to get the variable alone — and its answer is a whole range of values, not just one number. Solving works almost exactly like a one-step equation, with one new rule that trips up a lot of students, so we’ll make it impossible to forget. We’ll picture every answer on a number line.

The big idea

What Is a One-Step Inequality?

A one-step inequality is like a one-step equation but with an inequality sign — $<$, $>$, $\le$, or $\ge$ — instead of an equals sign. Solving it means finding every value of the variable that makes the statement true, so the answer is a range like $x > 5$, not just one number.

The one rule that’s different: when you multiply or divide both sides by a negative number, flip the inequality sign. (Adding or subtracting never flips it.)

How to solve a one-step inequality:

Undo the operation attached to the variable (add/subtract, or multiply/divide).
If you multiplied or divided by a negative, flip the sign.
Graph the solution on a number line.

Picturing the Answer on a Number Line

Open vs. closed

The solution $x \ge 1$

A closed (filled) circle means the endpoint is included ($\le$ or $\ge$); an open circle means it isn’t ($<$ or $>$). The arrow shows every number in the solution. Here $x \ge 1$ is a filled circle at 1 with the ray going right.

⚡ Solve an inequality

Solving, Step by Step

Add / subtract

No flip

Undo by the opposite operation.

$x – 5 > 2$
Add 5: $x >$ 7

Multiply / divide (positive)

No flip

Divide by a positive number — sign stays.

$4x \ge 20$
Divide by 4: $x \ge$ 5

Multiply / divide (negative)

Flip!

Multiplying by a negative reverses the order of the number line, so the sign flips.

$-3x \le 9$
Divide by $-3$, flip: $x \ge$ −3

Worked Examples

One operation, one move — then read the answer straight off the number line.

Example A — Subtract

Solve $x + 2 \ge 3$.

The variable has $+2$ attached, so subtract 2 from both sides: $x \ge 1$.
The sign is $\ge$, so the endpoint is included.
Graph it: closed circle at 1, arrow pointing right.

Answer: $x \ge 1$

Example B — Divide by a positive

Solve $3x \le 12$.

The variable is multiplied by 3, so divide both sides by 3: $x \le 4$.
Dividing by a positive, so the sign stays the same.
Graph it: closed circle at 4, arrow pointing left.

Answer: $x \le 4$

Example C — Divide by a negative (flip!)

Solve $-2x < 6$.

Divide both sides by $-2$ — dividing by a negative, so flip the sign: $x > -3$.
Quick check: $x = 0$ gives $-2(0) = 0 < 6$ ✓, and 0 is to the right of $-3$.
Graph it: open circle at $-3$, arrow pointing right.

Answer: $x > -3$

Example D — Negative coefficient

Solve $-x \ge 4$.

A bare $-x$ means $-1 \cdot x$, so divide both sides by $-1$ and flip: $x \le -4$.
The sign is $\le$, so the endpoint is included.
Graph it: closed circle at $-4$, arrow pointing left.

Answer: $x \le -4$

Inequalities in the Wild

Inequalities describe limits, not exact amounts. “You must be at least 48 inches to ride” is $h \ge 48$. “Stay under budget” with $50 and $5 items is $5x \le 50$, so $x \le 10$ items. Speed limits, minimum ages, weight capacities — train your ear for phrases like “at most,” “at least,” “no more than,” and “under” — they’re inequality signs in disguise, and once you spot the phrase, the symbol writes itself.

Don’t Forget to Flip

Forgetting the flip. The #1 inequality error: dividing or multiplying by a negative without reversing the sign. Circle the negative as a reminder.
Flipping when you shouldn’t. Adding or subtracting a negative does not flip the sign — only multiplying/dividing by a negative does.
Open vs. closed circle. Use a filled circle for $\le$/$\ge$ and an open circle for $<$/$>$. The endpoint is either included or it isn’t.
Graphing the arrow backwards. $x > -3$ points right (bigger numbers); $x < -3$ points left. Read the final sign, not the original.

Your Turn: Solve and Graph

Solve each, then say which way the number line points. Reveal to check.

$x + 4 > 9$
$x – 3 \le -1$
$5x < 20$
$-3x \ge 12$
$\dfrac{x}{2} > 4$
$-x + 1 < 5$
$-\dfrac{x}{2} \ge 3$

Show answers (with graphs)

$\color{blue}{x>5}$
$\color{blue}{x\le 2}$
$\color{blue}{x<4}$
$\color{blue}{x\le -4 \text{ (flipped)}}$
$\color{blue}{x>8}$
$\color{blue}{x>-4 \text{ (flipped)}}$
$\color{blue}{x\le -6 \text{ (flipped)}}$

1. $x > 5$

2. $x \le 2$

3. $x < 4$

4. $x \le -4$

5. $x > 8$

6. $x > -4$

7. $x \le -6$

Keep practicing

Make Your Own Inequalities Worksheet

Generate fresh one-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 ✏️ Quick Practice Drills

⚖️

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 — even adding a negative — never flips the sign.

What’s the difference between an open and closed circle?

An open circle means the endpoint is not included ($<$ or $>$); a closed, filled circle means it is included ($\le$ or $\ge$).

Why does an inequality have so many answers?

Because it asks for every value that makes the statement true, not just one. $x>5$ is satisfied by 6, 7, 100, and 5.0001 — the whole range to the right of 5.

What if the variable ends up on the right, like $7 < x$?

It’s the same solution, just written backwards. $7 < x$ means $x > 7$ — flip the whole statement (numbers, variable, and sign together) so the variable comes first.

How do I check an inequality solution?

Pick any number in your solution range and plug it into the original inequality; it should be true. Then test a number 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 nextHow to Solve Multi-Step InequalitiesContinue with Inequalities and Systems of EquationsOpen lesson →Up nextLinear InequalitiesContinue with Inequalities and Systems of EquationsOpen lesson →Up nextHow to Solve Systems of EquationsContinue with Inequalities and Systems of EquationsOpen lesson →

Effortless Math Team

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Effortless Math Team

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