Kenneth ChengMathematical modeling
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Properties of equalities and inequalities

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

Revisit first: Universal and existential quantifiers

TOPIC 01

Properties of equalities and inequalities

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

What you will be able to explain

  • Apply properties of equalities and inequalities with explicit conditions.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Order

Adding a common number preserves order; multiplying by a negative reverses it.

Legal transformations

Check divisors, signs and boundary cases.

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 properties of equalities and inequalities 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 and justify the sign direction.

−2x>6-2x>6
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Divide by a negative number.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    x<6/(−2)=−3x<6/(-2)=-3

  3. A negative multiplier reverses order.

x<-3.

Checks and common pitfalls: A negative multiplier reverses order.

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

Compare the reciprocals for 0<a<b.

0<a<b0<a<b
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Use a common positive denominator.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    1a−1b=b−aab>0\frac1a-\frac1b=\frac{b-a}{ab}>0

  3. The positivity assumptions justify the sign.

1/a>1/b.

Checks and common pitfalls: The positivity assumptions justify the sign.

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

Explain how the sign of a affects multiplying x<b by a.

x<bx<b
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Factor the difference of the proposed sides.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    ax−ab=a(x−b)ax-ab=a(x-b)

  3. The zero case is different from either strict order.

If a>0 then ax<ab; if a<0 then ax>ab; if a=0 the results are equal.

Checks and common pitfalls: The zero case is different from either strict order.

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 and justify the sign direction.

−3x>12-3x>12
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Divide by a negative number.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    x<12/(−3)=−4x<12/(-3)=-4

  3. A negative multiplier reverses order.

x<-4.

Checks and common pitfalls: A negative multiplier reverses order.

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

Given −2<x<5, find the interval containing −3x.

−2<x<5-2<x<5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Multiply every part by the negative number.
Hint 2
Both inequality signs reverse.
Worked solution

  1. Multiply every part by the negative number.

  2. Calculate or simplify this relation.

    6>−3x>−15  ⟺  −15<−3x<66>-3x>-15\iff-15<-3x<6

  3. Rewrite the final chain in increasing order.

−15<−3x<6.

Checks and common pitfalls: Rewrite the final chain in increasing order.

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

Find the least integer greater than every admissible sum a+b.

a<3, b<5a<3,\ b<5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Add the upper bounds.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    a+b<8a+b<8

  3. Sums approach this bound arbitrarily closely, so no smaller integer works.

The requested value is 8.

Checks and common pitfalls: Sums approach this bound arbitrarily closely, so no smaller integer works.

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

07 / Foundation#Your turn

Give a counterexample to squaring as a general order-preserving rule.

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

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Compare absolute values.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    (−4)2=16>9=(−3)2(-4)^2=16>9=(-3)^2

  3. Squaring is increasing only on nonnegative inputs.

Their squares satisfy 16>9.

Checks and common pitfalls: Squaring is increasing only on nonnegative inputs.

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.

08 / Standard#Your turn

Convert to a double 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
Track signs and legal operations before transforming an inequality.
Hint 2
Distance from 3 is less than three.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    −3<x−3<3⇒0<x<6-3<x-3<3\Rightarrow 0<x<6

  3. Both endpoints are excluded.

0<x<6.

Checks and common pitfalls: Both endpoints are excluded.

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

Does a<b and c<b guarantee a<c? Explain.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Choose a larger than c but still below b.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    a=5, c=4, b=6a=5,\ c=4,\ b=6

  3. Transitivity requires a chain, not merely a shared upper bound.

No; the two values below b may occur in either order.

Checks and common pitfalls: Transitivity requires a chain, not merely a shared upper bound.

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

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

  • Use the domain, units and sampling assumptions stated in the question.
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

  1. Subtraction reverses the comparison involving the subtracted terms.

  2. Calculate or simplify this relation.

    0<1,0<3,0−0=0>−2=1−30<1,\quad0<3,\quad0-0=0>-2=1-3

  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
  • 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 and justify the sign direction.

−4x>20-4x>20
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Divide by a negative number.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    x<20/(−4)=−5x<20/(-4)=-5

  3. A negative multiplier reverses order.

x<-5.

Checks and common pitfalls: A negative multiplier reverses order.

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

If a<0<b, compare 1/a and 1/b.

a<0<ba<0<b
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Compare the signs before invoking any monotonicity rule.
Hint 2
The reciprocal retains the sign of a nonzero number.
Worked solution

  1. Compare the signs before invoking any monotonicity rule.

  2. Calculate or simplify this relation.

    1/a<0<1/b1/a<0<1/b

  3. The reciprocal function is not decreasing across a domain interval containing its excluded zero.

1/a<1/b.

Checks and common pitfalls: The reciprocal function is not decreasing across a domain interval containing its excluded zero.

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

Find the least integer greater than every admissible sum a+b.

a<4, b<6a<4,\ b<6
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Track signs and legal operations before transforming an inequality.
Hint 2
Add the upper bounds.
Worked solution

  1. Track signs and legal operations before transforming an inequality.

  2. Calculate or simplify this relation.

    a+b<10a+b<10

  3. Sums approach this bound arbitrarily closely, so no smaller integer works.

The requested value is 10.

Checks and common pitfalls: Sums approach this bound arbitrarily closely, so no smaller integer works.

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 properties of equalities and inequalities?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • 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.

    Anticipated thinking

    • 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.

    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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