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Math lesson plan · 9th grade (Year 10 in England) · 45 minutes

Solve double inequalities like a < bx + c < d

Algebra

CCSS HSA.REI.B.3England: GCSE Higher Algebra

Objective and success criteria

Students will: Solve double inequalities like a < bx + c < d.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Compound inequalities and integer solutions; Solve multi-step inequalities, including reversing the sign

Leads to: the next skill in this area.

Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 5 min

  • Quick retrieval from earlier learning (Compound inequalities and integer solutions; Solve multi-step inequalities, including reversing the sign). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. Solve: 2x − 7 < −21 (x < −8 / x > −7 / x < 7 / x < −7) → x < −7Prompt: Solve it like an equation, step by step.
  2. Solve: −(x + 3) > x − 9 (x < 3 / x < −3 / x > 3 / x < 5) → x < 3Prompt: Expand the bracket, then collect x terms on one side.
  3. Solve: 4x + 8 < 36 (x > 7 / x < −7 / x < 6 / x < 7) → x < 7Prompt: Solve it like an equation, step by step.

Hook · 5 min

  • Show a "think of a number" puzzle and ask the class to work out the starting number.
  • Share the objective: Solve double inequalities like a < bx + c < d. Read the success criteria together.

I do: model it · 10 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: Solve: −13 < 4x − 1 < 7
  • Step 1: −12 < 4x < 8
  • Step 2: −3 < x < 2
  • Answer: −3 < x < 2
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 10 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. Solve: −27 < 4x − 3 < −11 (−7 < x < −1 / −6 > x > −2 / −9 < x < −5 / −6 < x < −2) → −6 < x < −2Prompt: Whatever you do, do it to all three parts.
  2. Solve: 10 < −4x + 6 < 26 (4 < x < 20 / −1 < x < −5 / −5 < x < −1 / 1 < x < 5) → −5 < x < −1Prompt: Subtract the constant from all three parts.

You do: independent practice · 10 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 5 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. Solve: 1 < −4x + 1 < 9 (−2 < x < 0 / 0 < x < 8 / 0 < x < 2 / 0 < x < −2) → −2 < x < 0Prompt: Subtract the constant from all three parts.
  2. Solve: 9 < 3x + 3 < 24 (2 < x < 7 / 2 > x > 7 / 1 < x < 8 / 6 < x < 21) → 2 < x < 7Prompt: Whatever you do, do it to all three parts.

Watch out for

  • Doing an operation to one side of an equation but not the other.
  • Reading 3x as 3 + x, or thinking x always stands for one particular number.

Support and stretch

Support

  • Review first: Compound inequalities and integer solutions.
  • Use concrete materials: balance scales or a balance drawing.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-9.

Free lesson plan from eurekami.com/lesson-plan-builder

eurekami

Solve double inequalities like a < bx + c < d

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  1. Question 1

    Solve: −17 < 4x − 1 < −9

    • A. −5 < x < −3
    • B. −16 < x < −8
    • C. −4 < x < −2
    • D. −4 > x > −2

  2. Question 2

    Solve: 2x > −6 and 2x ≤ −2. Write the answer as one double inequality.

    • A. −3 ≤ x < −1
    • B. −3 < x ≤ −1
    • C. x ≤ −3 or x ≥ −1
    • D. −3 ≤ x ≤ −1

  3. Question 3

    Solve: 10 < −4x + 2 < 26

    • A. 2 < x < 6
    • B. 8 < x < 24
    • C. −2 < x < −6
    • D. −6 < x < −2

  4. Question 4

    Solve: −6 < 5x − 6 < 29

    • A. 0 < x < 7
    • B. 0 > x > 7
    • C. 0 < x < 35
    • D. −1 < x < 8

  5. Question 5

    Solve: 3x − 2 > 1 and 3x − 2 ≤ 4. Write the answer as one double inequality.

    • A. 1 ≤ x < 2
    • B. 1 ≤ x ≤ 2
    • C. x ≤ 1 or x ≥ 2
    • D. 1 < x ≤ 2

  6. Question 6

    Solve: −8 < −x − 2 < −3

    • A. 6 < x < 1
    • B. −6 < x < −1
    • C. 1 < x < 6

  7. Question 7

    Solve: 4 < 5x + 4 < 34

    • A. 0 > x > 6
    • B. 0 < x < 6
    • C. −1 < x < 7
    • D. 4 < x < 10

  8. Question 8

    Solve: 3x − 4 ≥ −13 and 3x − 4 < −10. Write the answer as one double inequality.

    • A. −3 < x ≤ −2
    • B. −3 ≤ x ≤ −2
    • C. x ≤ −3 or x ≥ −2
    • D. −3 ≤ x < −2

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Answer key: Solve double inequalities like a < bx + c < d

  1. 1. C. −4 < x < −2

  2. 2. B. −3 < x ≤ −1

  3. 3. D. −6 < x < −2

  4. 4. A. 0 < x < 7

  5. 5. D. 1 < x ≤ 2

  6. 6. C. 1 < x < 6

  7. 7. B. 0 < x < 6

  8. 8. D. −3 ≤ x < −2