LEARN · EXPLAIN · REVISE

From a<b and c<d, may one conclude a−c<b−d? Give a counterexample.

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

高一必修第一册(A版).pdf · 2.1 · PDF 44 / printed page 37

TOPIC 01

Properties of equalities and inequalities

Build understanding of properties of equalities and inequalities through definitions, contrasting cases and justified applications.

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

01 / Standard#Your turn

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Subtraction reverses the comparison involving the subtracted terms.

Hint 2

Test a large gap between c and d.

Worked solution

Subtraction reverses the comparison involving the subtracted terms.
Calculate or simplify this relation.
$0<1,\quad0<3,\quad0-0=0>-2=1-3$
Inequalities may be added in the same direction, but not arbitrarily subtracted.

No; take a=0,b=1,c=0,d=3.

Checks and common pitfalls: Inequalities may be added in the same direction, but not arbitrarily subtracted.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

What must be true before using the main rule for properties of equalities and inequalities?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Apply properties of equalities and inequalities with explicit conditions.
Order: Adding a common number preserves order; multiplying by a negative reverses it.
Legal transformations: Check divisors, signs and boundary cases.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Adding a common number preserves order; multiplying by a negative reverses it.
Expected reasoning: Check divisors, signs and boundary cases.
Expected correction: Do not divide by a potentially zero expression.