Solving Two-Step Inequalities: The Mystery Number Challenge

Materials Needed

• Pencil and paper or a whiteboard
• Two different colored pencils or markers
• Optional: number line, calculator, or digital drawing tool
• Timer

Learning Objectives

By the end of this 15-minute lesson, the learner will be able to:

• Solve two-step inequalities using inverse operations.
• Reverse the inequality symbol when multiplying or dividing by a negative number.
• Represent a solution on a number line.
• Check whether a possible value makes an inequality true.

Success Criteria

I can:

• Undo addition or subtraction first.
• Undo multiplication or division second.
• Reverse the inequality sign when dividing or multiplying by a negative number.
• Show my answer with an inequality and, when asked, on a number line.

Introduction: The Mystery Number Challenge — 2 Minutes

Hook: Imagine a secret number. When you multiply it by 3 and add 4, the result is less than 19. What numbers could the secret number be?

Today, you will become an inequality detective. Unlike an equation, which usually has one answer, an inequality can have a whole group of possible answers.

Quick prediction: Ask yourself: “Would 4 work?” Test it mentally:

3(4) + 4 = 16, and 16 is less than 19, so 4 works.

Tell the learner: “First I’ll model the strategy, then we’ll solve one together, and finally you’ll solve some independently.”

Body: I Do — Teacher/Parent Models — 4 Minutes

Key Idea

To solve a two-step inequality, undo the operations in reverse order:

1. Undo addition or subtraction.
2. Undo multiplication or division.
3. Reverse the inequality symbol if multiplying or dividing by a negative number.

Example 1: No Sign Change

Solve:

3x + 4 < 19

1. Undo +4 by subtracting 4 from both sides:
   3x < 15
2. Undo multiplication by 3 by dividing both sides by 3:
   x < 5

Check: Choose x = 4. Then 3(4) + 4 = 16, and 16 < 19, so the answer makes sense.

Number line: Use an open circle at 5 because 5 is not included. Shade to the left because the solutions are numbers less than 5.

Example 2: Negative Division

Solve:

-2x + 6 ≥ 14

1. Undo +6 by subtracting 6 from both sides:
   -2x ≥ 8
2. Divide both sides by -2. Because we divide by a negative number, reverse the inequality symbol:
   x ≤ -4

Memory tip: “When negatives flip the sign, the answer changes direction.”

Body: We Do — Solve Together — 4 Minutes

Work through each problem with the learner. Ask the learner to explain what operation should be undone first and why.

Problem 1

2x - 5 ≤ 9

1. Add 5 to both sides:
   2x ≤ 14
2. Divide by 2:
   x ≤ 7

Discussion question: Should the inequality sign flip? No. We divided by a positive number.

Problem 2

-3x - 2 > 10

1. Add 2 to both sides:
   -3x > 12
2. Divide by -3 and reverse the sign:
   x < -4

Quick check: Ask the learner to test x = -5:

-3(-5) - 2 = 13, and 13 > 10, so -5 works.

Think-Pair-Share Alternative

If another person is available, the learner explains each step to that person. If working independently, the learner can explain the steps aloud to a stuffed animal, recording, or imaginary “math detective partner.”

Body: You Do — Independent Math Detective Mission — 3 Minutes

Solve each inequality. Show your steps and identify whether the inequality sign changes.

1. 4x + 3 < 19
2. 5x - 7 ≥ 18
3. -2x + 1 ≤ 9

Optional challenge: Create your own two-step inequality with a solution of x > 6. Then solve it to check that your creation works.

Answer Key

1. 4x + 3 < 19
   4x < 16
   x < 4
2. 5x - 7 ≥ 18
   5x ≥ 25
   x ≥ 5
3. -2x + 1 ≤ 9
   -2x ≤ 8
   Divide by -2 and reverse the sign:
   x ≥ -4

Conclusion: Recap and Exit Ticket — 2 Minutes

Ask the learner to complete this sentence:

“To solve a two-step inequality, I first ________, then ________. If I divide or multiply by a negative number, I ________.”

Expected response: “I undo addition or subtraction first, then undo multiplication or division. I reverse the inequality sign.”

Exit Ticket

Solve:

-4x + 8 > 20

Answer:

-4x > 12

Divide by -4 and reverse the sign:

x < -3

Assessment

• Formative assessment: Listen for correct identification of the first operation to undo and ask whether the sign should flip.
• Practice assessment: Check the learner’s three independent solutions for accurate steps and answers.
• Summative assessment: Use the exit ticket. The learner demonstrates mastery by correctly solving the inequality, reversing the sign, and explaining why.

Differentiation and Adaptations

• Support: Use a two-column organizer labeled “Operation” and “Undo Operation.” Highlight the number attached to the variable.
• Visual support: Draw a number line for every answer and use an open circle for < or > and a closed circle for ≤ or ≥.
• Language support: Provide the sentence frame: “I undo ___ by ___ on both sides.”
• Hands-on option: Write each step on separate cards and physically arrange the cards in the correct order.
• Extension: Ask the learner to write an inequality for a real-life situation, such as budgeting, game points, or distance traveled.
• Real-world connection: Example: “You have $20 and want to save at least $5. If each snack costs $3, how many snacks can you buy?” This can be represented as 3x + 5 ≤ 20, giving x ≤ 5.