不等式bùděngshì

Core Concept

An inequality is a mathematical statement that uses inequality symbols (>, <, ≥, ≤) to express the relationship between two numbers or expressions.

Basic Inequality Symbols

• $a > b$: $a$ is greater than $b$
• $a < b$: $a$ is less than $b$
• $a \geq b$: $a$ is greater than or equal to $b$
• $a \leq b$: $a$ is less than or equal to $b$
• $a \neq b$: $a$ is not equal to $b$

Properties of Inequalities

1. Transitivity

$a > b, b > c \Rightarrow a > c$

2. Addition Property

$a > b \Rightarrow a + c > b + c$

3. Multiplication Property

• When $c > 0$: $a > b \Rightarrow ac > bc$
• When $c < 0$: $a > b \Rightarrow ac < bc$ (inequality reverses!)

Common Types of Inequalities

Linear Inequality

$2x + 3 > 7 \Rightarrow x > 2$

Quadratic Inequality

$x^2 - 5x + 6 < 0 \Rightarrow 2 < x < 3$

Rational Inequality

$\frac{x-1}{x+2} > 0 \Rightarrow x < -2 \text{ or } x > 1$

Absolute Value Inequality

$|x| < a \Rightarrow -a < x < a \quad (a > 0)$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★★☆☆☆)

Solve the inequality: $3x - 5 < 7$

Solution: $3x < 12$ $x < 4$

Answer: $x < 4$ or $(-\infty, 4)$

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Example 2: Intermediate (Difficulty ★★★☆☆)

Solve the inequality: $x^2 - 3x - 4 \leq 0$

Solution:

Step 1: Factor $x^2 - 3x - 4 = (x-4)(x+1)$

Step 2: Find critical points $x = -1$ or $x = 4$

Step 3: Number line method

• When $x < -1$: $(x-4)(x+1) > 0$
• When $-1 \leq x \leq 4$: $(x-4)(x+1) \leq 0$ ✓
• When $x > 4$: $(x-4)(x+1) > 0$

Answer: $-1 \leq x \leq 4$ or $[-1, 4]$

Common Mistakes

❌ Mistake 1: Forgetting to reverse when multiplying by negative

Wrong: $-2x > 4$, divide both sides by $-2$ to get $x > -2$ ✗

Correct: $-2x > 4$, divide both sides by $-2$ to get $x < -2$ ✓

❌ Mistake 2: Absolute value inequality error

Wrong: $|x| > 2$ has solution $-2 < x < 2$ ✗

Correct: $|x| > 2$ has solution $x < -2$ or $x > 2$ ✓

Study Tips

1. ✅ Master properties, especially reversing when multiplying/dividing by negatives
2. ✅ Use number line method for quadratic inequalities
3. ✅ Convert rational inequalities to polynomial form
4. ✅ Use case analysis for absolute value inequalities

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💡 Exam Tip: Inequalities are essential CSCA content, accounting for about 25% of algebra problems. Master all types!