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Week 3: Rational Functions, Inequalities, and Modulus

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Week 3: Rational Functions, Inequalities, and Modulus

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Course: Jan 2026 - Mathematics I Difficulty: Advanced Setup Focus: Algebraic manipulation, solving inequalities, and absolute value behavior.

1. Rational Functions

A Rational Function

R(x)

is a function that can be expressed as the ratio of two polynomials,

P(x)

and

Q(x)

R(x) = \frac{P(x)}{Q(x)}

where

Q(x)

is not the zero polynomial.

1.1 Domain of a Rational Function

Since division by zero is undefined, the domain of

R(x)

is the set of all real numbers except those values of

x

for which

Q(x) = 0

. Example: The domain of

f(x) = \frac{x+2}{x^2 - 4}

is all real numbers except

x = 2

and

x = -2

1.2 Asymptotes

An asymptote is a line or curve that the graph of a function approaches as it moves towards infinity.

Vertical Asymptotes: Occur at the zeroes of the denominator $Q(x)$ that are NOT zeroes of the numerator $P(x)$ .
Horizontal Asymptotes: Describe the end behavior as $x \to \pm \infty$ $x \to \pm \infty$ . They depend on the degrees of $P(x)$ $P (x)$ (let's call it $n$ $n$ ) and $Q(x)$ $Q (x)$ (call it $m$ $m$ ).
- If $n < m$ , horizontal asymptote is $y = 0$ (the $x$ -axis).
- If $n = m$ , horizontal asymptote is $y = \frac{a_n}{b_m}$ (ratio of leading coefficients).
- If $n > m$ , there is no horizontal asymptote.

2. Inequalities

Inequalities express the relative size or order of two values (

<, >, \le, \ge

2.1 Linear Inequalities

Solved similarly to linear equations, with one crucial rule: Multiplying or dividing both sides by a negative number flips the inequality symbol.

-2x < 6 \implies x > -3

2.2 Quadratic and Polynomial Inequalities

To solve inequalities like

ax^2 + bx + c > 0

Find the critical points: These are the roots of the corresponding equation $ax^2 + bx + c = 0$ .
Number Line Method (Wavy Curve Method):
- Plot the distinct real roots on a number line. This divides the line into intervals.
- Pick a test point in each interval to evaluate the sign of the polynomial, or use the "right-most interval is positive" rule if the leading coefficient is positive, alternating signs if multiplicities are odd.
- Select the intervals that satisfy the inequality ( $> 0$ means positive, $< 0$ means negative).

2.3 Rational Inequalities

For

\frac{P(x)}{Q(x)} \ge 0

Find critical values where $P(x) = 0$ (numerator roots) and where $Q(x) = 0$ (denominator roots/vertical asymptotes).
Plot all these points on a number line.
Use test intervals to determine signs.
Important: Points where $Q(x) = 0$ are NEVER included in the solution set, even if the inequality is $\le$ or $\ge$ , because division by zero is undefined.

3. Modulus (Absolute Value) Function

The absolute value of a real number

x

, denoted as

|x|

, is its distance from zero on the number line. It is always non-negative.

x & \text{if } x \ge 0 \ -x & \text{if } x < 0 \end{cases}$$ ### 3.1 Properties of Modulus 1. $|a| \ge 0$ 2. $|ab| = |a||b|$ 3. $|\frac{a}{b}| = \frac{|a|}{|b|}$ for $b \neq 0$ 4. $|a|^2 = a^2$ 5. **Triangle Inequality:** $|a + b| \le |a| + |b|$ ### 3.2 Solving Modulus Equations and Inequalities * **Equation:** $|x| = a$ (where $a \ge 0$) implies $x = a$ or $x = -a$. * **Less Than:** $|x| < a$ (where $a > 0$) translates to the compound inequality $-a < x < a$. Geometrically, the distance is strictly less than $a$. * **Greater Than:** $|x| > a$ (where $a > 0$) translates to $x > a$ or $x < -a$. Expected values are outside the interval $[-a, a]$. > [!TIP] > Always verify solutions for modulus equations involving variables outside the modulus (e.g., $|2x - 3| = x + 1$) by plugging them back into the original equation to check for extraneous solutions. --- ### 🧭 Navigation - **📘 Textbook Notes:** [Week 3 Notes](./Jan%202026%20-%20Mathematics%20I%20-%20Week%203%20-%20Textbook%20Notes.md) - **📝 Assignment:** [Week 3 Assignment](./Jan%202026%20-%20Mathematics%20I%20-%20Week%203%20-%20Graded%20Assignment%203.md) - **⬅️ Previous Week:** [Week 2 Notes](./Jan%202026%20-%20Mathematics%20I%20-%20Week%202%20-%20Textbook%20Notes.md) - **➡️ Next Week:** [Week 4 Notes](./Jan%202026%20-%20Mathematics%20I%20-%20Week%204%20-%20Textbook%20Notes.md) ---