Kenneth ChengMathematical modeling
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Solve the inequality with a negative right side.

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

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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.

Try it. Leave your reasoning visible.

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

01 / 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.

Focus on one question

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.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗