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Rational Inequalities

CSCA Rational Inequalities study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Rational Inequalities

Rational inequalities are a common source of errors in the CSCA exam. The main traps are **division by zero** and **multiplying by the denominator blindly**.

1. Core Method: Equivalent Transformation

The most efficient way is not case analysis, but transforming the rational inequality into a **Polynomial Inequality**.

#### Transformation Rules

* **Type 1: Strict Inequalities ($>, <$)**

Based on "same signs positive, different signs negative":

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

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

**Note**: Just turn division into multiplication.

* **Type 2: Non-strict Inequalities ($\ge, \le$)**

You MUST explicitly ensure the **denominator is not zero**:

$$\frac{f(x)}{g(x)} \ge 0 \iff \begin{cases} f(x) \cdot g(x) \ge 0 \\ g(x) \neq 0 \end{cases}$$

2. Number Line / Sign Chart Method

For complex rational inequalities, using a Number Line (Sign Chart) is intuitive.

**Steps**:

1. **Find Points**: Find **Zeros** (Numerator = 0) and **Poles** (Denominator = 0).

2. **Plot Points**:

* **Solid Dot**: Zeros when the inequality includes $\ge$ or $\le$.

* **Hollow Circle**: Zeros for strict inequalities, and **ALL Poles (denominator)**.

3. **Determine Signs**: Check the sign of the expression in each interval (Test Points or Wavy Curve).

3. The Fatal Trap: DO NOT Cross-Multiply

**Warning**: When solving $\frac{1}{x} > 1$, **NEVER** multiply both sides by $x$ to get $1 > x$!

**Reason**: The sign of $x$ is unknown. If $x < 0$, the inequality sign must flip. Cross-multiplying leads to missing or incorrect solutions.

**Correct Method**: Move terms and combine fractions. $\frac{1}{x} - 1 > 0 \Rightarrow \frac{1-x}{x} > 0$.

4. Worked Examples

**Ex 1: Standard Type**

Solve $\frac{x-1}{x+2} > 0$.

**Sol**:

Equivalent to $(x-1)(x+2) > 0$.

Roots are $-2, 1$. Parabola opens up.

Take the "outside" intervals.

**Ans**: $(-\infty, -2) \cup (1, +\infty)$.

**Ex 2: Inclusive Inequality with Rearranging**

Solve $\frac{2x}{x-1} \le 1$.

**Sol**:

1. **Rearrange** (Do not multiply by $x-1$):

$$\frac{2x}{x-1} - 1 \le 0 \Rightarrow \frac{2x - (x-1)}{x-1} \le 0 \Rightarrow \frac{x+1}{x-1} \le 0$$

2. **Transform**:

$$\begin{cases} (x+1)(x-1) \le 0 \\ x-1 \neq 0 \end{cases}$$

3. **Solve**:

From $(x+1)(x-1) \le 0$, we get $[-1, 1]$.

Exclude $x=1$ (denominator).

> **Ans**: $[-1, 1)$.

**Ex 3: Multiple Factors**

Solve $\frac{x^2-4}{x-3} \le 0$.

**Sol**:

Factorize: $\frac{(x-2)(x+2)}{x-3} \le 0$.

* **Zeros** (Numerator): $x=2, x=-2$ (Solid, since $\le$).

* **Poles** (Denominator): $x=3$ (Hollow, always).

* **Sign Check**:

* $x > 3$: (+)

* $2 < x < 3$: (-)

* $-2 < x < 2$: (+)

* $x < -2$: (-)

Take regions $\le 0$.

**Ans**: $(-\infty, -2] \cup [2, 3)$.