Two-Step Inequalities

Learn to solve inequalities that require two operations, combining addition/subtraction with multiplication/division.

Intermediate30 minLesson
Printable worksheet

Definition

A two-step inequality requires two operations to solve. Just like two-step equations, we use inverse operations in reverse order.
Strategy: Undo addition/subtraction FIRST, then undo multiplication/division.
2x+5>112x + 5 > 11
Step 1: Subtract 5 from both sides: 2x>62x > 6
Step 2: Divide both sides by 2: x>3x > 3
Remember the FLIP rule: When multiplying or dividing by a negative number, flip the inequality sign!
−3x+7≤16-3x + 7 \leq 16
Step 1: Subtract 7: −3x≤9-3x \leq 9
Step 2: Divide by −3-3 (FLIP!): x≥−3x \geq -3

Try it now

Solve: 2x+3>112x + 3 > 11

Worked Examples

Solve: 3x+7>223x + 7 > 22
1

Identify the operations on x

xx is multiplied by 3, then 7 is added → Operations: ×3, then +7
2

Undo addition first

Subtract 7 from both sides: 3x+7−7>22−73x + 7 - 7 > 22 - 7 → 3x>153x > 15
3

Undo multiplication

Divide both sides by 3: 3x3>153\frac{3x}{3} > \frac{15}{3} → x>5x > 5
4

Check with a value

Try x=6x = 6: 3(6)+7=253(6) + 7 = 25, and 25>2225 > 22 ✓ → Solution verified

Common Mistakes

Undoing operations in the wrong order

Why it's wrong: Students try to divide first instead of handling addition/subtraction first.

Correct: Always undo addition/subtraction BEFORE multiplication/division. Think PEMDAS in reverse!

Forgetting to flip when dividing by a negative

Why it's wrong: After multiple steps, students lose track of whether they're dividing by positive or negative.

Correct: Circle the coefficient! If it's negative and you're dividing, put a big FLIP reminder.

Only applying operations to one side

Why it's wrong: With two steps, students sometimes forget to apply the second operation to both sides.

Correct: Whatever you do to the left side, you MUST do to the right side. Check both sides each step.

Distributing incorrectly with negative signs

Why it's wrong: In problems like 5−4x≥175 - 4x \geq 17, students mishandle the negative coefficient.

Correct: Keep track of signs carefully. 5−4x5 - 4x means +5+5 and −4x-4x. The coefficient of xx is −4-4.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Solve Linear Inequalities | Step-by-Step with Graphs and Interval Notation

by Understand The Math

Watch on YouTube

Interactive Visual

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x > 5·(5, ∞)

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x ≤ -3·(−∞, -3]

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

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Practice Problems

18 problems
Problem 1 of 18
Easy
Solve: 2x+3>112x + 3 > 11

Why It Matters

Two-step inequalities appear constantly in real-life planning and decision-making:
  • Budgeting: If a phone plan costs 15 euros per month plus a 50 euro activation fee, and you can spend at most 200 euros this year, how many months can you keep it?
  • Distance and travel: If you've already traveled 30 km and your car uses 8 liters per 100 km, with 20 liters of fuel left, how much further can you go?
  • Business: A company has 1000 euros fixed costs plus 5 euros per item. To make profit, revenue must exceed costs.
  • Grades: If your current average is 72% based on 4 tests, what score do you need on the 5th test to reach at least 75%?
Mastering two-step inequalities is essential for solving complex real-world problems!

Real World Applications

Phone Plan Budget

Mobile plans often have a fixed monthly fee plus charges for extra data or minutes.

Example:

A plan costs 25 euros per month plus 3 euros per GB of extra data. If you can spend at most 40 euros: 25+3g≤4025 + 3g \leq 40, so g≤5g \leq 5 GB.

1Try It Yourself

Your phone plan costs 20 euros monthly. Each additional text costs 0.10 euros. You have a budget of at most 35 euros for this month.

How many extra texts can you send?

Step 1: Choose the right answer

Choose the inequality for texts tt:

Saving for a Purchase

When you want to buy something but also need to keep a minimum balance.

Example:

You have 180 euros in savings. After weekly expenses of 12 euros, you want at least 60 euros left. How many weeks until you must stop? 180−12w≥60180 - 12w \geq 60.

2Try It Yourself

You have 250 euros saved. Each week you spend 15 euros on lunch. You want to keep at least 100 euros for emergencies.

For how many weeks can you afford this lunch budget?

Step 1: Choose the right answer

Choose the inequality:

Temperature Requirements

Industrial processes often require temperatures within specific ranges.

Example:

A chemical must be heated from 20°C, rising 5°C per minute, but must stay below 80°C. How long can you heat it? 20+5t<8020 + 5t < 80, so t<12t < 12 minutes.

3Try It Yourself

A freezer starts at 10°C and cools at a rate of 3°C per minute. Food safety requires the temperature to reach at most -8°C.

After how many minutes will the freezer be cold enough?

Step 1: Choose the right answer

Choose the inequality:

Key Takeaways

  • 1Two-step inequalities require two inverse operations to solve
  • 2Order matters: Undo addition/subtraction FIRST, then multiplication/division
  • 3Work in reverse order of PEMDAS (SADMEP)
  • 4CRITICAL: When multiplying or dividing by a NEGATIVE number, FLIP the inequality sign
  • 5Always check your solution by substituting a value from the solution set

Frequently Asked Questions

One-step inequalities need only one operation (like x+5>8x + 5 > 8). Two-step inequalities need two operations in sequence (like 2x+5>112x + 5 > 11).
One-step inequalities need only one operation (like x+5>8x + 5 > 8). Two-step inequalities need two operations in sequence (like 2x+5>112x + 5 > 11).
No! Only flip when you MULTIPLY or DIVIDE by a negative number. Adding or subtracting negatives does NOT require a flip.
When you isolate xx by dividing both sides by −3-3, you MUST flip the inequality sign because −3-3 is negative.
Technically yes, but it usually makes things harder. Following the standard order (undo +/− first, then ×/÷) is more reliable.

Glossary

Two-step inequality
An inequality that requires two inverse operations to solve (e.g., 2x+5>112x + 5 > 11)
Inverse operations
Operations that undo each other: addition/subtraction and multiplication/division
Coefficient
The number multiplied by a variable (e.g., in −3x-3x, the coefficient is −3-3)
Solution set
All values that make the inequality true

Formula Card

Standard Form

ax+b>cax + b > c

Two-step inequality with variable term first

Step 1: Isolate Variable Term

ax+b−b>c−b⇒ax>c−bax + b - b > c - b \Rightarrow ax > c - b

Undo addition/subtraction first

Step 2: Solve for Variable

axa>c−ba\frac{ax}{a} > \frac{c-b}{a} or FLIP if a<0a < 0

Divide by coefficient, flip if negative

Flip Rule Summary

Only flip when × or ÷ by negative

Adding/subtracting never requires a flip

Today's challengeOne problem, three tries

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