Modulus & Inequalities - Formulas & Rules
IB Mathematics Analysis & Approaches (SL & HL)
📐 Modulus (Absolute Value) Definition
Definition:
The modulus (or absolute value) of a real number gives the positive distance from zero.
\[|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\]
Alternative Notation:
\[|x| = \sqrt{x^2}\]
Key Properties:
Domain: All real numbers \(\mathbb{R}\)
Range: Non-negative real numbers \([0, \infty)\)
🔑 Properties of Modulus
Non-negativity:
\[|x| \geq 0 \quad \text{for all } x \in \mathbb{R}\]
Zero Property:
\[|x| = 0 \iff x = 0\]
Symmetry:
\[|-x| = |x|\]
Product Property:
\[|xy| = |x| \cdot |y|\]
Quotient Property:
\[\left|\frac{x}{y}\right| = \frac{|x|}{|y|} \quad (y \neq 0)\]
Triangle Inequality:
\[|x + y| \leq |x| + |y|\]
Reverse Triangle Inequality:
\[|x - y| \geq ||x| - |y||\]
📊 Modulus Function Graphs
Basic Form:
\[f(x) = |x|\]
• V-shaped graph with vertex at origin \((0, 0)\)
• Continuous everywhere but not differentiable at \(x = 0\)
• Symmetric about the y-axis (even function)
General Transformed Form:
\[f(x) = a|x - h| + k\]
• \(a\): Vertical stretch/compression (negative \(a\) reflects in x-axis)
• \(h\): Horizontal shift (vertex moves to \(x = h\))
• \(k\): Vertical shift (vertex moves to \(y = k\))
• Vertex: Located at \((h, k)\)
🔍 Solving Modulus Equations
Basic Equation Form:
\[|x| = a \quad (a \geq 0)\]
Solution: \(x = a\) or \(x = -a\)
If \(a < 0\): No solution (modulus is always non-negative)
General Approach for \(|f(x)| = a\):
Step 1: Split into two equations:
• \(f(x) = a\)
• \(f(x) = -a\)
Step 2: Solve both equations
Step 3: Check solutions in the original equation
Equations with Two Moduli:
For \(|f(x)| = |g(x)|\):
Solve: \(f(x) = g(x)\) or \(f(x) = -g(x)\)
📉 Solving Modulus Inequalities
Type 1: "Less Than" Inequalities
\[|x| < a \quad (a > 0)\]
\[\text{Solution: } -a < x < a\]
Type 2: "Greater Than" Inequalities
\[|x| > a \quad (a > 0)\]
\[\text{Solution: } x < -a \text{ or } x > a\]
General Form: \(|f(x)| < a\)
\[-a < f(x) < a\]
General Form: \(|f(x)| > a\)
\[f(x) < -a \text{ or } f(x) > a\]
📏 Linear Inequalities
Standard Form:
\[ax + b < c \quad \text{(or } \leq, >, \geq\text{)}\]
Solve like equations but remember: when multiplying or dividing by a negative number, reverse the inequality sign
Important Rule:
\[\text{If } a < b \text{ then } -a > -b\]
Multiplying by \(-1\) reverses the inequality
📊 Quadratic Inequalities
Standard Form:
\[ax^2 + bx + c < 0 \quad \text{(or } \leq, >, \geq\text{)}\]
Solution Method:
Step 1: Solve the corresponding equation \(ax^2 + bx + c = 0\)
Step 2: Sketch the parabola or use a sign table
Step 3: Determine intervals where the inequality is satisfied
• For \(< 0\) or \(\leq 0\): Find where graph is below x-axis
• For \(> 0\) or \(\geq 0\): Find where graph is above x-axis
🔢 Rational Inequalities
Standard Form:
\[\frac{f(x)}{g(x)} < 0 \quad \text{(or } \leq, >, \geq\text{)}\]
Solution Method:
Step 1: Find zeros of numerator (\(f(x) = 0\))
Step 2: Find zeros of denominator (\(g(x) = 0\)) - these are vertical asymptotes
Step 3: Create sign table with all critical values
Step 4: Test each interval
Important: Never include values where denominator = 0
📈 Polynomial Inequalities (HL)
Standard Form:
\[P(x) < 0 \quad \text{(or } \leq, >, \geq\text{)}\]
Solution Method:
Step 1: Find all roots of \(P(x) = 0\)
Step 2: Create sign diagram with all roots
Step 3: Consider multiplicity of roots:
• Odd multiplicity: Sign changes across the root
• Even multiplicity: Sign stays the same
Step 4: Identify intervals satisfying the inequality
⚖️ Key Inequality Rules
Transitive Property:
\[\text{If } a < b \text{ and } b < c, \text{ then } a < c\]
Addition Property:
\[\text{If } a < b, \text{ then } a + c < b + c\]
Multiplication Property (Positive):
\[\text{If } a < b \text{ and } c > 0, \text{ then } ac < bc\]
Multiplication Property (Negative):
\[\text{If } a < b \text{ and } c < 0, \text{ then } ac > bc\]
⚠️ Inequality sign reverses!
Reciprocal Property:
\[\text{If } 0 < a < b, \text{ then } \frac{1}{a} > \frac{1}{b}\]
⚠️ Inequality sign reverses!
💡 Exam Tip: For modulus inequalities, always sketch the graphs or use sign tables. Remember that \(|x| < a\) gives a bounded interval while \(|x| > a\) gives two unbounded intervals. When solving inequalities, be careful with multiplication/division by negative numbers - always reverse the inequality sign. Use your GDC to verify solutions graphically.