【PEP】Junior High School Math, Grade 7, Semester 2: From Balance to Asymmetry – Properties of Inequalities and Sign Reversal

In this lesson, we’ll move from the 'aesthetic of balance' in equations to the 'dynamic asymmetry' of inequalities. The core idea is understanding when the inequality sign maintains its 'inertia' and when it undergoes a 'dramatic reversal'—specifically, how multiplying by a negative number (Property 3) breaks the original order. This is the foundation for mastering compound inequality operations.

1. Difference Comparison Method: The Essence of Inequality

The essence of inequality lies in the relative displacement of numbers on the number line. This approach—using the result of subtraction to determine size—is the fundamental logic behind solving complex inequalities:

When $a - b > 0$, then $a > b$;
When $a - b = 0$, then $a = b$;
When $a - b < 0$, then $a < b$.

2. Sign Preservation: Translation and Positive Scaling

Follow Property 1 and Property 2 of inequalities. When adding or subtracting the same number from both sides, or multiplying or dividing both sides by a positive number, the points on the number line shift or scale, but their relative order remains unchanged.

Property 1: Add (or subtract) the same number (or expression) to both sides of an inequality—the direction of the inequality sign remains unchanged.
Property 2: Multiply (or divide by) both sides of an inequality by the same positive number—the direction of the inequality sign remains unchanged.

3. Mirror Effect: The 'Singularity' of Sign Reversal

This is the most critical concept in this lesson. When multiplying (or dividing) both sides of an inequality by the same negative number, the direction of the inequality signmust reverse. This reveals the 'mirror flip' effect of negative signs in inequality operations.

Property 3 (Core)

If $a > b$ and $c < 0$, then $ac < bc$ (or $\frac{a}{c} < \frac{b}{c}$).

🎯 Core Formula Summary

1. If $a > b$, then $a \pm c > b \pm c$.
2. If $a > b$ and $c > 0$, then $ac > bc$.
3. If $a > b$ and $c < 0$, then $ac < bc$.

QUESTION 1

Fill in the blanks with "$>$" or "$<$": $5 > 3$, $5+2 \_\_ 3+2$, $5-2 \_\_ 3-2$.

>, >

<, <

>, <

<, >

QUESTION 2

Given $6 > 2$, then $6 \times (-5) \_\_ 2 \times (-5)$.

$>$

$<$

$=$

Cannot be determined

QUESTION 3

Given $a > b$, then $-4a \_\_ -4b$.

$>$

$<$

≥

≤

QUESTION 4

Compare the sizes of $\sqrt{8}$ and $\sqrt{10}$.

$\sqrt{8} > \sqrt{10}$

$\sqrt{8} < \sqrt{10}$

$\sqrt{8} = \sqrt{10}$

QUESTION 5

Zhao Jun says the inequality $a > 2a$ can never hold. Is he correct?

Yes, because $1 > 2$ is false

No, it holds when $a < 0$

Yes, derived from Property 2

No, it holds when $a = 0$

QUESTION 6

Compare the sizes of $\frac{\sqrt{5}-1}{2}$ and $0.5$.

$>$

$<$

$=$

QUESTION 7

Solve the inequality $x - 7 > 26$.

$x > 19$

$x < 33$

$x > 33$

$x > 33$ and $x \neq 0$

QUESTION 8

Solve the inequality $-4x > 3$.

$x > -\frac{3}{4}$

$x < -\frac{3}{4}$

$x < \frac{3}{4}$

$x > \frac{4}{3}$

QUESTION 9

A rectangular container is 5 cm long, 3 cm wide, and 10 cm high. The initial water level is 3 cm. Water is injected at $V\text{cm}^3$. What is the range of $V$?

$0 < V < 150$

$0 < V \leq 105$

$3 < V < 10$

$V \leq 150$

QUESTION 10

In a competition with 20 questions, you earn 10 points for each correct answer, lose 5 points for each wrong or unanswered question. To score over 90 points, what is the minimum number of correct answers?