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Math lesson plan · 9th grade (Year 10 in England) · 60 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 · 6 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 − 5 ≤ 11 (x ≥ −9 / x ≤ −8 / x ≥ −8 / x ≥ 8) → x ≥ −8Prompt: Solve it like an equation, step by step.
  2. Solve: (x + 4) ≤ −3x + 16 (x ≤ 5 / x ≥ 3 / x ≤ −3 / x ≤ 3) → x ≤ 3Prompt: Expand the bracket, then collect x terms on one side.
  3. Solve: −x − 3 > 4 (x < −8 / x < 7 / x < −7 / x > −7) → x < −7Prompt: Solve it like an equation, step by step.

Hook · 6 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 · 12 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 · 12 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: 4 < 5x − 6 < 14 (−4 < x < −2 / 2 > x > 4 / 10 < x < 20 / 2 < x < 4) → 2 < x < 4Prompt: Whatever you do, do it to all three parts.
  2. Solve: −4 < −4x − 4 < 20 (0 < x < −6 / 0 < x < 24 / −6 < x < 0 / 0 < x < 6) → −6 < x < 0Prompt: Subtract the constant from all three parts.

You do: independent practice · 17 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 · 7 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. Solve: −14 < −4x + 2 < 6 (−1 < x < 4 / −16 < x < 4 / 4 < x < −1 / −4 < x < 1) → −1 < x < 4Prompt: Subtract the constant from all three parts.
  2. Solve: −19 < 5x + 1 < 11 (−20 < x < 10 / −4 < x < 2 / −4 > x > 2 / −3 < x < 3) → −4 < x < 2Prompt: 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: 2x − 2 > −6 and 2x − 2 ≤ 2. Write the answer as one double inequality.

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

  2. Question 2

    Solve: 2 < −2x + 4 < 10

    • A. −1 < x < 3
    • B. −2 < x < 6
    • C. 1 < x < −3
    • D. −3 < x < 1

  3. Question 3

    Solve: 14 < 4x + 6 < 26

    • A. 8 < x < 20
    • B. 1 < x < 6
    • C. 2 > x > 5
    • D. 2 < x < 5

  4. Question 4

    Solve: 2x − 2 ≥ −6 and 2x − 2 < −2. Write the answer as one double inequality.

    • A. −2 < x ≤ 0
    • B. −2 ≤ x < 0
    • C. x ≤ −2 or x ≥ 0
    • D. −2 ≤ x ≤ 0

  5. Question 5

    Solve: −9 < −x − 4 < −6

    • A. 5 < x < 2
    • B. −5 < x < −2
    • C. 2 < x < 5

  6. Question 6

    Solve: −9 < 2x + 1 < 1

    • A. −4 < x < 1
    • B. −5 > x > 0
    • C. −5 < x < 0
    • D. −6 < x < 1

  7. Question 7

    Solve: 2x + 3 ≥ 7 and 2x + 3 < 11. Write the answer as one double inequality.

    • A. 2 ≤ x < 4
    • B. 2 < x ≤ 4
    • C. 2 ≤ x ≤ 4
    • D. x ≤ 2 or x ≥ 4

  8. Question 8

    Solve: −26 < −4x + 2 < −2

    • A. −28 < x < −4
    • B. −7 < x < −1
    • C. 1 < x < 7
    • D. 7 < x < 1

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

  1. 1. D. −2 < x ≤ 2

  2. 2. D. −3 < x < 1

  3. 3. D. 2 < x < 5

  4. 4. B. −2 ≤ x < 0

  5. 5. C. 2 < x < 5

  6. 6. C. −5 < x < 0

  7. 7. A. 2 ≤ x < 4

  8. 8. C. 1 < x < 7