How to Solve One-Step Inequalities? (+FREE Worksheet!)
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.
Quick answer: A one-step inequality needs only one inverse operation to get the variable alone. Do the same thing to both sides, just like a one-step equation. The one new rule: if you multiply or divide both sides by a negative number, flip the inequality sign. Example: −3x > 12. Divide both sides by −3 and flip the sign to get x < −4.
Solve One-Step Inequalities: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
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
- Divide both sides by -3.
- Reverse the inequality sign.
- Simplify 12 divided by -3.
Keep the sign
- Subtract 5 from both sides.
- No negative multiplication or division happened.
- Keep the sign direction.
Try one before moving on
Solve One-Step Inequalities: pop-up practice
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.
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.
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
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 inequalitySolving, Step by Step
No flip
Undo by the opposite operation.
Add 5: \(x >\) 7
No flip
Divide by a positive number, sign stays.
Divide by 4: \(x \ge\) 5
Flip!
Multiplying by a negative reverses the order of the number line, so the sign flips.
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)
- \(\color{blue}{x>5}\)
- \(\color{blue}{x\le 2}\)
- \(\color{blue}{x<4}\)
- \(\color{blue}{x\le -4 \text{ (flipped)}}\)
- \(\color{blue}{x>8}\)
- \(\color{blue}{x>-4 \text{ (flipped)}}\)
- \(\color{blue}{x\le -6 \text{ (flipped)}}\)
Make Your Own Inequalities Worksheet
Generate fresh one-step inequalities with a full answer key, print or save as a PDF.
One example for each operation
Every one-step inequality undoes a single operation. This table shows one of each, so you can see when the sign stays and when it flips.
| Inequality | Inverse step | Answer | Flip the sign? |
|---|---|---|---|
| x + 6 < 10 | Subtract 6 from both sides | x < 4 | No |
| x − 2 ≥ −5 | Add 2 to both sides | x ≥ −3 | No |
| 4x ≤ 28 | Divide both sides by 4 | x ≤ 7 | No |
| x/(−5) > 2 | Multiply both sides by −5 | x < −10 | Yes |
Notice that adding or subtracting never flips the sign, even with negative numbers. Only multiplying or dividing by a negative does.
Turning word phrases into inequality symbols
Word problems hide the symbol inside a phrase. Learn these pairs:
- at least, no less than, minimum: ≥
- at most, no more than, maximum: ≤
- more than, greater than: >
- less than, fewer than: <
Example 1: Maria has saved $25 and needs at least $60. How much more money, m, does she need? Write 25 + m ≥ 60. Subtract 25 from both sides: m ≥ 35. She needs $35 or more.
Example 2: Movie tickets cost $8 each, and you can spend no more than $56. How many tickets, t, can you buy? Write 8t ≤ 56. Divide both sides by 8: t ≤ 7. You can buy up to 7 tickets.
Frequently Asked Questions
What is a one-step inequality in math?
A one-step inequality is an inequality you can solve with a single inverse operation, such as adding, subtracting, multiplying or dividing both sides by the same number. For example, x − 7 ≥ −2 becomes x ≥ 5 after adding 7 to both sides. The answer is a set of values, not just one number, so every number 5 or greater works.
How do you solve one-step inequalities by multiplying or dividing?
Multiply or divide both sides by the same number, and flip the inequality sign only if that number is negative. For −5x ≥ 20, divide both sides by −5 and flip the sign to get x ≤ −4. For x/2 ≤ 3, multiply by positive 2 and keep the sign: x ≤ 6. Test a value to check, such as x = −5 in the first one.
How do you graph one-step inequalities on a number line?
Solve first, then mark the boundary number. Use a closed (filled) circle for ≤ or ≥ because the number is included, and an open circle when the sign is strictly less than or greater than. Shade toward the values that work. For x ≥ −2, put a closed circle at −2 and shade to the right.
How do you solve one-step inequalities with fractions?
Multiply both sides by the reciprocal of the fraction next to the variable. For (2/5)x ≤ 4, multiply both sides by 5/2 to get x ≤ 10. For x/3 ≥ 2, multiply by 3 to get x ≥ 6. The sign stays the same because 5/2 and 3 are positive; it would flip only if you multiplied by a negative fraction.
How do you solve one-step inequality word problems?
Choose a variable, translate the key phrase into a symbol, and solve. “At most” means ≤ and “at least” means ≥. If you have $20 and each ticket costs $4, then 4t ≤ 20, so t ≤ 5: you can buy at most 5 tickets. Always check that the answer makes sense, since you cannot buy part of a ticket.
What is a two-step inequality?
A two-step inequality needs two inverse operations, usually undoing addition or subtraction first and then multiplication or division. For 2x + 3 ≤ 11, subtract 3 to get 2x ≤ 8, then divide by 2 to get x ≤ 4. The flip rule is the same as for one-step inequalities: reverse the sign only when you multiply or divide by a negative.
Do you flip the sign when adding or subtracting a negative number?
No. Adding or subtracting any number, positive or negative, keeps the sign the same. For example, x + (−4) > 3 becomes x > 7 after you add 4 to both sides. The sign only flips when you multiply or divide both sides by a negative number.
What if the variable has a negative coefficient on the right side, like 12 ≤ −4x?
Divide both sides by −4 and flip the sign: 12 ÷ (−4) ≥ x, so −3 ≥ x. Read it from the variable’s side and it means x ≤ −3. Test x = −4: −4 × −4 = 16, and 12 ≤ 16 is true, so the answer checks out.
Do you flip the sign if the answer is negative but you divided by a positive number?
No. What matters is the number you divide by, not the answer. In 3x > −12, you divide by positive 3, so the sign stays: x > −4. A negative number on the other side of the inequality does not change the rule.
How do you write the answer to a one-step inequality in interval notation?
Use parentheses for < and > and brackets for ≤ and ≥. Infinity always gets a parenthesis. So x < −4 becomes (−∞, −4), and x ≥ 1 becomes [1, ∞). The bracket matches the closed circle on a number line, and the parenthesis matches the open circle.
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