One-Step Inequalities

Learn to solve inequalities that require just one operation, and understand why the inequality sign flips when multiplying or dividing by a negative.

Intermediate25 minLesson
Printable worksheet

Definition

A one-step inequality is an inequality that can be solved with a single operation (addition, subtraction, multiplication, or division).
Key Principle: Solve inequalities the same way you solve equations - do the same thing to both sides!
x+5>8⇒x>3x + 5 > 8 \quad \Rightarrow \quad x > 3
CRITICAL RULE: When you multiply or divide both sides by a negative number, you must FLIP the inequality sign!
−2x>6⇒x<−3-2x > 6 \quad \Rightarrow \quad x < -3
OriginalAfter × or ÷ by negative
<<>>
>><<
≤\leq≥\geq
≥\geq≤\leq

Try it now

Solve: x+5>9x + 5 > 9

Worked Examples

Solve: x+7>12x + 7 > 12
1

Identify the operation

7 is being added to xx → Operation: addition
2

Use inverse operation

Subtract 7 from both sides: x+7−7>12−7x + 7 - 7 > 12 - 7 → x>5x > 5
3

Check with a value

Try x=6x = 6: 6+7=136 + 7 = 13, and 13>1213 > 12 ✓ → Solution verified

Common Mistakes

Forgetting to flip the sign when dividing by a negative

Why it's wrong: This is the most common error! Students apply inverse operations correctly but forget this special rule.

Correct: ALWAYS check: Am I multiplying or dividing by a NEGATIVE? If yes, FLIP the inequality sign!

Flipping the sign for any operation with negatives

Why it's wrong: Students over-apply the rule, flipping when adding or subtracting negative numbers too.

Correct: Only flip when MULTIPLYING or DIVIDING by a negative. Adding x+(−5)>3x + (-5) > 3 does NOT require a flip!

Doing different operations to each side

Why it's wrong: Unlike equations, the visual difference in inequalities can confuse students.

Correct: Just like equations: whatever you do to the left side, you MUST do to the right side too.

Writing the answer in the wrong direction

Why it's wrong: After solving, students write 5>x5 > x instead of x<5x < 5.

Correct: Convention is to write the variable on the left: x<5x < 5 is clearer than 5>x5 > x.

Interactive Visual

to

x ≥ 7·[7, ∞)

Click on numbers to select them. Adjust the range to explore different values.

to

x < -5·(−∞, -5)

Click on numbers to select them. Adjust the range to explore different values.

Interactive Sandbox

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Practice Problems

16 problems
Problem 1 of 16
Easy
Solve: x+5>9x + 5 > 9

Why It Matters

One-step inequalities are the foundation for solving more complex inequalities and appear everywhere in real life:
  • Budgeting: If you have 50 euros and need to save at least 20, how much can you spend? (50−x≥2050 - x \geq 20)
  • Time management: If you need at least 8 hours of sleep and wake at 7 AM, when must you be asleep by? (7−t≥87 - t \geq 8)
  • Sports: How many more points does a team need to win? (score+x>opponent\text{score} + x > \text{opponent})
  • Cooking: If a recipe serves 4 and you need to feed at least 12, how many batches? (4n≥124n \geq 12)
Mastering one-step inequalities prepares you for multi-step inequalities and real-world problem solving!

Real World Applications

Budgeting and Saving

When you need to save a minimum amount, inequalities help determine how much you can spend.

Example:

You have 75 euros and want to keep at least 30 euros for emergencies. How much can you spend? 75−x≥3075 - x \geq 30, so x≤45x \leq 45.

1Try It Yourself

You earn 12 euros per hour at your part-time job. You want to earn at least 60 euros this week.

How many hours must you work?

Step 1: Choose the right answer

Which inequality fits the hours hh?

Temperature Limits

Science experiments often require temperatures within certain ranges.

Example:

A chemical reaction requires the temperature to stay below 100°C. If the room is 25°C and heat is added at 15° per minute: 25+15t<10025 + 15t < 100.

2Try It Yourself

A freezer must keep food at or below −18°C-18°C. The current temperature is −10°C-10°C, and it drops 2°C2°C per hour.

After how many hours will the freezer reach the required temperature?

Step 1: Choose the right answer

After hh hours the temperature is −10−2h-10 - 2h. Which inequality fits?

Sports and Games

Determining what scores are needed to win or qualify.

Example:

Your team has 48 points. Each basket is worth 2 points. How many more baskets to reach 60? 48+2b≥6048 + 2b \geq 60, so b≥6b \geq 6.

3Try It Yourself

A diver needs a score of at least 50 points to qualify. Each dive is scored out of 10, and she has completed 4 dives with an average of 8.5.

What average score does she need on her remaining 2 dives?

Step 1: Choose the right answer

If her total from 4 dives is 4×8.5=344 \times 8.5 = 34, which inequality fits?

Key Takeaways

  • 1Solve one-step inequalities using inverse operations, just like equations
  • 2Addition and subtraction: simply apply the inverse operation
  • 3Multiplication and division by POSITIVE numbers: apply inverse operation normally
  • 4CRITICAL: When multiplying or dividing by a NEGATIVE number, FLIP the inequality sign
  • 5Always check your answer by substituting a value from the solution set

Frequently Asked Questions

When you multiply numbers by −1-1, their order reverses. For example, 3>23 > 2 but −3<−2-3 < -2. The flip keeps the inequality TRUE after the operation.
When you multiply numbers by −1-1, their order reverses. For example, 3>23 > 2 but −3<−2-3 < -2. The flip keeps the inequality TRUE after the operation.
No! Only flip when MULTIPLYING or DIVIDING by a negative. Adding −5-5 to both sides does not flip anything.
Pick any number from your solution set and plug it into the original inequality. If it works, you're correct! Also try a number NOT in the solution set to confirm it fails.
You can solve it normally, then rewrite with the variable on the left. Remember: 5>x5 > x is the same as x<5x < 5.

Glossary

One-step inequality
An inequality that can be solved with a single operation (add, subtract, multiply, or divide)
Inverse operation
The operation that undoes another: addition undoes subtraction, multiplication undoes division
Solution set
All values that make the inequality true
Flip the sign
Changing the direction of an inequality (e.g., >> becomes <<) when multiplying or dividing by a negative

Formula Card

Addition/Subtraction Rule

x+a>b⇒x>b−ax + a > b \Rightarrow x > b - a

Subtract $a$ from both sides (no flip)

Positive Multiplication

ax>b⇒x>baax > b \Rightarrow x > \frac{b}{a} (when a>0a > 0)

Divide both sides by $a$ (no flip)

Negative Multiplication

ax>b⇒x<baax > b \Rightarrow x < \frac{b}{a} (when a<0a < 0)

Divide by $a$ AND FLIP the sign!

Sign Flip Rule

−x>c⇒x<−c-x > c \Rightarrow x < -c

Multiply by $-1$ and flip the inequality

Today's challengeOne problem, three tries

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