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.

Tutor-style math help

Solve One-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 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

2 weeks ago

Effortless Math Team

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