Kenneth ChengMathematical modeling
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Absolute values and solution sets

Read the idea, work independently, then explain what changed.

2027 JM01 考試大綱 · 7. Algebraic inequalities · PDF 2 / printed page 2

Revisit first: Basic operations on setsQuadratic functions, equations and inequalities

TOPIC 01

Absolute values and solution sets

Build understanding of absolute values and solution sets through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Choose an equivalent method and explain exclusions before calculating.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Distance

|x−a| measures the distance between x and a on the real line.

Logic of constraints

Simultaneous constraints use intersection; alternative cases use union.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in absolute values and solution sets changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

The difference is 2ab=4. Equality holds exactly when ab=0.(a+b)² = 9a²+b² = 5

The difference is 2ab=4. Equality holds exactly when ab=0.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Solve the strict absolute-value inequality.

∣x−2∣<3|x-2|<3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
A distance less than 3 stays inside the two bounds.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    −3<x−2<3-3<x-2<3

  3. Strict bounds exclude endpoints.

(-1,5).

Checks and common pitfalls: Strict bounds exclude endpoints.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Solve the exterior distance inequality.

∣x−2∣≥2|x-2|\ge2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
The point may lie on either exterior side.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    x−2≤−2orx−2≥2x-2\le-2\quad\text{or}\quad x-2\ge2

  3. Use a union, since a point need not satisfy both inequalities.

x≤0 or x≥4.

Checks and common pitfalls: Use a union, since a point need not satisfy both inequalities.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Solve by comparing distances to 0 and 2.

∣x∣<∣x−2∣|x|<|x-2|
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
Both sides are nonnegative, so squaring is reversible here.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    x2<(x−2)2  ⟺  4x<4x^2<(x-2)^2\iff 4x<4

  3. The midpoint is excluded because the comparison is strict.

x<2/2.

Checks and common pitfalls: The midpoint is excluded because the comparison is strict.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Solve the strict absolute-value inequality.

∣x−3∣<3|x-3|<3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
A distance less than 3 stays inside the two bounds.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    −3<x−3<3-3<x-3<3

  3. Strict bounds exclude endpoints.

(0,6).

Checks and common pitfalls: Strict bounds exclude endpoints.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Solve the exterior distance inequality.

∣x−3∣≥2|x-3|\ge2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
The point may lie on either exterior side.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    x−3≤−2orx−3≥2x-3\le-2\quad\text{or}\quad x-3\ge2

  3. Use a union, since a point need not satisfy both inequalities.

x≤1 or x≥5.

Checks and common pitfalls: Use a union, since a point need not satisfy both inequalities.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Solve the simultaneous inequalities.

∣x−3∣≤3;x>3|x-3|\le3;\quad x>3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
Keep points satisfying both constraints.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    [0,6]∩(3,∞)=(3,6][0,6]\cap(3,\infty)=(3,6]

  3. Use intersection for simultaneous requirements.

(3,6].

Checks and common pitfalls: Use intersection for simultaneous requirements.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Count the integer solutions.

∣x−3∣<2;x∈Z|x-3|<2;\quad x\in\mathbb Z
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
List the integers strictly between the endpoints.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    x∈{2,3,4}x\in\{2,3,4\}

  3. Continuous interval length is not the number of integer points.

The requested value is 3.

Checks and common pitfalls: Continuous interval length is not the number of integer points.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Solve the inequality with a negative right side.

∣x−3∣≤−1|x-3|\le-1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
Absolute value is always nonnegative.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    ∣x−3∣≥0>−1|x-3|\ge0>-1

  3. Squaring would erase the crucial sign condition.

No solution.

Checks and common pitfalls: Squaring would erase the crucial sign condition.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Find the minimum of the sum of distances and every point attaining it.

∣x∣+∣x−3∣|x|+|x-3|
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
Use the triangle inequality, then inspect points between the centres.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    ∣x∣+∣x−3∣≥∣x−(x−3)∣=3|x|+|x-3|\ge|x-(x-3)|=3

  3. There is an interval of minimisers, rather than only its midpoint.

Minimum 3, attained on [0,3].

Checks and common pitfalls: There is an interval of minimisers, rather than only its midpoint.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Solve by comparing distances to 0 and 3.

∣x∣<∣x−3∣|x|<|x-3|
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
Both sides are nonnegative, so squaring is reversible here.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    x2<(x−3)2  ⟺  6x<9x^2<(x-3)^2\iff 6x<9

  3. The midpoint is excluded because the comparison is strict.

x<3/2.

Checks and common pitfalls: The midpoint is excluded because the comparison is strict.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Solve the strict absolute-value inequality.

∣x−4∣<3|x-4|<3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
A distance less than 3 stays inside the two bounds.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    −3<x−4<3-3<x-4<3

  3. Strict bounds exclude endpoints.

(1,7).

Checks and common pitfalls: Strict bounds exclude endpoints.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Solve the exterior distance inequality.

∣x−4∣≥2|x-4|\ge2
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
The point may lie on either exterior side.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    x−4≤−2orx−4≥2x-4\le-2\quad\text{or}\quad x-4\ge2

  3. Use a union, since a point need not satisfy both inequalities.

x≤2 or x≥6.

Checks and common pitfalls: Use a union, since a point need not satisfy both inequalities.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Solve the simultaneous inequalities.

∣x−4∣≤3;x>4|x-4|\le3;\quad x>4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Interpret absolute value as distance and use intersections or unions correctly.
Hint 2
Keep points satisfying both constraints.
Worked solution

  1. Interpret absolute value as distance and use intersections or unions correctly.

  2. Calculate or simplify this relation.

    [1,7]∩(4,∞)=(4,7][1,7]\cap(4,\infty)=(4,7]

  3. Use intersection for simultaneous requirements.

(4,7].

Checks and common pitfalls: Use intersection for simultaneous requirements.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for absolute values and solution sets?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Choose an equivalent method and explain exclusions before calculating.
    • Distance: |x−a| measures the distance between x and a on the real line.
    • Logic of constraints: Simultaneous constraints use intersection; alternative cases use union.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: |x−a| measures the distance between x and a on the real line.
    • Expected reasoning: Simultaneous constraints use intersection; alternative cases use union.
    • Expected correction: A candidate produced by algebra may violate the original domain or sign condition.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗