Kenneth ChengMathematical modeling
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Prove this universal statement by cases.

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

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高一必修 第一册(A版).pdf · 1.5 · PDF 33 / printed page 26

Revisit first: Sufficient and necessary conditions

TOPIC 01

Universal and existential quantifiers

Build understanding of universal and existential quantifiers through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Apply universal and existential quantifiers with explicit conditions.
  • 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 / Transfer#Your turn

Prove this universal statement by cases.

∀x∈R, ∣x∣≥x\forall x\in\mathbb R,\ |x|\ge x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the definition of absolute value.
Hint 2
Separate nonnegative and negative inputs.
Worked solution

  1. Use the definition of absolute value.

  2. Calculate or simplify this relation.

    x≥0⇒∣x∣=xx\ge0\Rightarrow |x|=x

  3. Calculate or simplify this relation.

    x<0⇒∣x∣=−x>xx<0\Rightarrow |x|=-x>x

  4. The two cases cover the complete real domain.

True for all real x.

Checks and common pitfalls: The two cases cover the complete real domain.

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 universal and existential quantifiers?
  • Which representation makes this task easier, and why?
  • Change one assumption. Does the conclusion survive?

Board plan

  • Apply universal and existential quantifiers with explicit conditions.
  • Scope: Universal means every domain member; existential means at least one witness.
  • Negation: Swap the quantifier and negate the predicate while keeping the domain.
  • Close with: conditions → representation → reasoning → check.

Anticipated thinking

  • Expected reasoning: Universal means every domain member; existential means at least one witness.
  • Expected reasoning: Swap the quantifier and negate the predicate while keeping the domain.
  • Expected correction: Examples alone do not prove a universal statement.

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 ↗