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Teacher Guide: 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.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Inequalities. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Solve one-step inequalities involving addition and subtraction
  • Solve one-step inequalities involving multiplication and division
  • Understand and apply the rule for flipping the inequality sign
  • Verify solutions by substituting values into the original inequality
  • Translate real-world problems into one-step inequalities and solve them
Prerequisites
  • • Understanding of inequality symbols (<<, >>, ≤\leq, ≥\geq)
  • • Solving one-step equations
  • • Operations with positive and negative numbers
  • • Understanding of inverse operations
Discussion Starters
  • 1. Why do you think the inequality sign flips when we multiply by a negative?
  • 2. Can you think of a real-life situation where you need to know a minimum or maximum value?
  • 3. What's the difference between solving 2x=102x = 10 and 2x>102x > 10?
  • 4. If x>5x > 5 and y<5y < 5, what can you say about xx and yy?
Common Misconceptions

The sign always flips when there's a negative number anywhere

Remediation: Emphasize: ONLY flip when you MULTIPLY or DIVIDE by a negative. Work through examples like x−5>3x - 5 > 3 (no flip) vs −x>3-x > 3 (flip).

Thinking x>−5x > -5 means xx is negative

Remediation: Use the number line: x>−5x > -5 includes 0, 1, 2, etc. Show that positive numbers ARE greater than −5-5.

Forgetting that inequalities have infinite solutions

Remediation: Unlike equations with one answer, x>3x > 3 has infinitely many solutions: 4, 5, 100, 3.5, etc. Use number line visualization.

Differentiation Ideas

For Struggling Students:

  • • Start with positive coefficients only before introducing negative multiplication
  • • Use physical balance scale manipulatives
  • • Create a 'Flip Checklist': Is it multiply/divide? Is it by a negative? If BOTH yes, flip!
  • • Practice checking answers immediately after solving

For On-Level Students:

  • • Solve all types of one-step inequalities
  • • Practice identifying when to flip the sign without prompts
  • • Translate word problems into inequalities

For Advanced Students:

  • • Challenge with variables on the right side
  • • Introduce compound inequalities preview
  • • Prove why the sign flips using number line reasoning
  • • Create their own real-world inequality problems
Standards Alignment
  • 7.EE.B.4b (CCSS.MATH.CONTENT.7.EE.B.4.B)

    Solve word problems leading to inequalities and graph the solution set

  • 6.EE.B.8 (CCSS.MATH.CONTENT.6.EE.B.8)

    Write an inequality to represent a constraint or condition in a real-world problem

  • 6.EE.B.5 (CCSS.MATH.CONTENT.6.EE.B.5)

    Understand solving an inequality as finding values that make it true

Lesson Resources
  • visualBalance Scale Interactive

    See how operations affect both sides of an inequality

  • activityFlip or No Flip Game

    Practice identifying when to flip the inequality sign

  • worksheetReal-World Inequality Problems

    Translate situations into inequalities and solve

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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

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.

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

Why do we flip the inequality when multiplying by a negative?

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.

Do I flip for adding or subtracting negative numbers?

No! Only flip when MULTIPLYING or DIVIDING by a negative. Adding −5-5 to both sides does not flip anything.

How do I check my answer?

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.

What if the variable is on the right side?

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

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