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Basic Properties and Solutions of Inequalities

CSCA Basic Properties and Solutions of Inequalities study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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International students preparing for CSCA Math, Physics, Chemistry, or Chinese exams.

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Basic Properties and Solutions of Inequalities

Inequalities are a core algebraic tool in the CSCA exam. Solving an inequality essentially involves transforming complex expressions into the form $x > a$ or $x < a$ using specific properties.

1. Three Key Properties

* **Transitivity**: If $a < b$ and $b < c$, then $a < c$.

* **Addition Property**: Adding (or subtracting) the same number to both sides **does not change** the direction of the inequality.

* $a < b \iff a+c < b+c$

* **Multiplication Property —— 🚩 The Only Trap!**

* Multiplying by a **positive** number: Direction stays the **same**. ($a < b, c > 0 \implies ac < bc$)

* Multiplying by a **negative** number: Direction **reverses**. ($a < b, c < 0 \implies ac > bc$)

2. General Solution Methods

#### (1) Linear Inequalities

* **Method**: Rearrange terms $\rightarrow$ Combine like terms $\rightarrow$ Divide by the coefficient.

* **Note**: Remember to **flip the inequality sign** if dividing by a negative coefficient.

#### (2) Quadratic Inequalities

Form: $ax^2+bx+c > 0 (a>0)$.

* **Steps**:

1. **Find Roots**: Solve $ax^2+bx+c=0$ for roots $x_1, x_2$ (assume $x_1 < x_2$).

2. **Sketch Graph**: Draw a parabola opening upwards.

3. **Determine Intervals**:

* For $>0$ (Graph above x-axis): Take the "tails" (outside), i.e., $x < x_1$ or $x > x_2$.

* For $<0$ (Graph below x-axis): Take the "middle" (between), i.e., $x_1 < x < x_2$.

#### (3) Rational Inequalities

Form: $\frac{f(x)}{g(x)} > 0$.

* **Core Trick**: Use the principle that "same signs give positive, different signs give negative" to convert division into multiplication.

* $\frac{f(x)}{g(x)} > 0 \iff f(x) \cdot g(x) > 0$

* $\frac{f(x)}{g(x)} \le 0 \iff f(x) \cdot g(x) \le 0$ AND $g(x) \neq 0$ (denominator cannot be zero).

3. Exam Tips

* For higher-degree inequalities, the **Method of Intervals (Wavy Curve Method)** is a powerful tool to master in advanced modules.

* Never cross-multiply the denominator in rational inequalities (since its sign is unknown). Instead, move terms to one side or use the multiplication conversion method above.