Inequality Techniques

🎢 Advanced Inequality Techniques: Your Superhero Toolkit!

Imagine you’re a detective solving mysteries. But instead of finding criminals, you’re finding which numbers make math sentences true. Today, we’re learning 6 super-powered tools that help us solve tricky inequality puzzles!

🌊 The Wavy Curve Method (The Rollercoaster Trick)

The Story

Picture a rollercoaster that goes up and down. The wavy curve method is like drawing that rollercoaster on paper!

When you have an inequality like:

(x - 2)(x + 3)(x - 5) > 0

You’re asking: “Where is this rollercoaster above the ground (positive)?”

How It Works

Step 1: Find the “zero points” (where the rollercoaster touches ground)

• x = 2, x = -3, x = 5

Step 2: Put them in order on a number line

←---(-3)----(2)----(5)---→

Step 3: Draw a wavy curve starting from the top right

Step 4: The curve alternates: above, below, above, below…

graph TD
    A["Find zeros: -3, 2, 5"] --> B["Put on number line"]
    B --> C["Draw wave from top right"]
    C --> D["Above ground = positive"]
    D --> E["Below ground = negative"]

Example

Solve: (x - 1)(x + 2) > 0

• Zeros: x = 1 and x = -2
• Wave pattern: +, -, +
• Answer: x < -2 OR x > 1 ✓

🎯 Think of it as: A snake slithering across your number line!

📊 Logarithmic Inequalities (The Secret Code Solver)

The Story

Remember when you learned that log is the opposite of powers? Like how subtraction undoes addition?

If 2³ = 8, then log₂(8) = 3

Logarithmic inequalities are like solving coded messages!

The Golden Rule

When the base is BIGGER than 1: The inequality sign STAYS the same.

When the base is SMALLER than 1: The inequality sign FLIPS!

graph TD
    A["Check the base"] --> B{Base > 1?}
    B -->|Yes| C["Keep the sign"]
    B -->|No| D["Flip the sign"]

Example 1: Base > 1

Solve: log₂(x) > 3

• Base is 2 (bigger than 1), so sign stays
• x > 2³
• x > 8 ✓

Example 2: Base < 1

Solve: log₀.₅(x) > 2

• Base is 0.5 (smaller than 1), so sign FLIPS
• x < (0.5)²
• x < 0.25 (but x must be positive!)
• Answer: 0 < x < 0.25 ✓

⚠️ Never forget: You can only take log of positive numbers!

🚀 Exponential Inequalities (The Growth Detective)

The Story

Exponential means things that grow SUPER fast—like how one rabbit becomes millions!

When solving exponential inequalities, the base is your best friend (or enemy).

The Golden Rule (Same as logarithms!)

Base > 1: Sign stays the same. Base < 1 (but positive): Sign flips!

Example 1: Base > 1

Solve: 2ˣ > 8

• 2ˣ > 2³
• Base is 2 (bigger than 1)
• x > 3 ✓

Example 2: Base < 1

Solve: (1/3)ˣ > 9

• (1/3)ˣ > (1/3)⁻²
• Base is 1/3 (smaller than 1), so FLIP!
• x < -2 ✓

graph TD
    A["Exponential Inequality"] --> B["Make same base"]
    B --> C{Base > 1?}
    C -->|Yes| D["Keep inequality"]
    C -->|No| E["Flip inequality"]

🧠 Memory trick: Small bases are “rebels”—they flip everything!

📏 Modulus Inequalities (The Distance Measurer)

The Story

Modulus (or absolute value) is just distance from zero. It’s always positive!

Think of it like this: Whether you walk 5 steps left OR 5 steps right, you still walked 5 steps!

|−5| = 5 and |5| = 5

The Two Types

Type 1: |x| < a (Less than)

• Means: x is BETWEEN -a and a
• -a < x < a

Type 2: |x| > a (Greater than)

• Means: x is OUTSIDE the range
• x < -a OR x > a

Example 1: Less Than

Solve: |x - 3| < 5

• -5 < (x - 3) < 5
• -5 + 3 < x < 5 + 3
• -2 < x < 8 ✓

Example 2: Greater Than

Solve: |2x + 1| > 7

• 2x + 1 < -7 OR 2x + 1 > 7
• 2x < -8 OR 2x > 6
• x < -4 OR x > 3 ✓

graph TD
    A["Modulus Inequality"] --> B{Less than or Greater than?}
    B -->|< a| C["Sandwich: -a &lt; stuff &lt; a"]
    B -->|> a| D["Split: stuff &lt; -a OR stuff &gt; a"]

🎯 Remember: “Less than” = stay INSIDE, “Greater than” = go OUTSIDE!

⚖️ AM-GM Inequality (The Average Superpower)

The Story

AM-GM is like a magic rule that says:

“The normal average is ALWAYS bigger than (or equal to) the special multiplication average!”

AM = Arithmetic Mean (add and divide) GM = Geometric Mean (multiply and take root)

The Formula

For positive numbers a and b:

(a + b)/2 ≥ √(ab)

In words: Average ≥ Square root of product

When Are They Equal?

Only when a = b!

Example 1: Finding Minimum

Find the minimum value of x + 4/x for x > 0

Using AM-GM:

• (x + 4/x)/2 ≥ √(x × 4/x)
• (x + 4/x)/2 ≥ √4
• (x + 4/x)/2 ≥ 2
• x + 4/x ≥ 4 ✓

Minimum value is 4 (when x = 2)

Example 2: Three Numbers

For positive a, b, c:

(a + b + c)/3 ≥ ∛(abc)

graph TD
    A["AM-GM Inequality"] --> B["AM ≥ GM always"]
    B --> C["Equal only when all numbers are same"]
    C --> D["Use to find min/max values"]

💡 Pro tip: AM-GM helps find minimum sums or maximum products!

🔗 Cauchy-Schwarz Inequality (The Power Pair)

The Story

This inequality is like a superhero team-up! It tells us something amazing about pairs of numbers.

Imagine you have two teams:

• Team A: a₁, a₂
• Team B: b₁, b₂

Cauchy-Schwarz says there’s a special limit to how their combinations can behave!

The Formula

(a₁b₁ + a₂b₂)² ≤ (a₁² + a₂²)(b₁² + b₂²)

In simple words: The mixed product squared is never bigger than the product of sums of squares!

When Are They Equal?

When the teams are proportional: a₁/b₁ = a₂/b₂

Example

Prove: (3×4 + 4×3)² ≤ (3² + 4²)(4² + 3²)

Left side: (12 + 12)² = 24² = 576

Right side: (9 + 16)(16 + 9) = 25 × 25 = 625

576 ≤ 625 ✓

Useful Form

If x + y = S (constant), find minimum of x² + y²:

By Cauchy-Schwarz:

• (x + y)² ≤ 2(x² + y²)
• S² ≤ 2(x² + y²)
• x² + y² ≥ S²/2

Minimum is S²/2 when x = y = S/2

graph TD
    A["Cauchy-Schwarz"] --> B["Links sums and products"]
    B --> C["Equality when proportional"]
    C --> D["Great for optimization problems"]

🌟 Big idea: This inequality connects different types of sums in a beautiful way!

🎯 Quick Summary: Your 6 Supertools

🏆 You Did It!

You now have six powerful techniques to solve advanced inequalities. Each one is like a different key that opens different types of locks.

Remember: Practice makes perfect! The more problems you solve, the faster you’ll recognize which tool to use.

🚀 You’re not just learning math—you’re becoming a problem-solving superhero!