Inequalities: The Art of “Not Exactly Equal”

🎭 The Story of the Velvet Rope

Imagine you’re at a fancy party. There’s a velvet rope at the door. The sign says: “You must be at least 13 years old to enter.”

This isn’t about being exactly 13. It’s about being 13 or more. That’s an inequality!

Equations are like saying: “You must be exactly 5 feet tall to ride this ride.” Inequalities are like saying: “You must be at least 4 feet tall to ride this ride.”

One is strict. The other gives you room.

📖 What is an Inequality?

An inequality is a math statement that compares two things and says one is:

• Greater than another
• Less than another
• Greater than or equal to another
• Less than or equal to another
• Not equal to another

Think of it like this:

Your piggy bank has some coins. Your friend’s piggy bank has some coins too.

• If you have MORE coins → Your coins > Friend’s coins
• If you have FEWER coins → Your coins < Friend’s coins
• If you have the SAME → Your coins = Friend’s coins (this is an equation!)

🌟 Real-Life Inequalities

✏️ Inequality Notation

We use special symbols to write inequalities. Think of them as hungry alligators — they always want to eat the bigger number!

🔍 Examples

5 > 3     →  "5 is greater than 3"
2 < 7     →  "2 is less than 7"
x ≥ 10    →  "x is 10 or more"
y ≤ 5     →  "y is 5 or less"
z ≠ 0     →  "z is anything except 0"

💡 Memory Trick

The pointy end always points to the smaller number! The open mouth always faces the bigger number!

BIG > small     small < BIG

📊 Interval Notation

Interval notation is a shortcut way to write a range of numbers.

Imagine a number line as a highway. Interval notation tells you where to start and where to stop.

The Symbols

🎯 Examples

🧠 Remember

• Parentheses ( ) = “Don’t touch this number!”
• Brackets [ ] = “This number is included!”
• Infinity ∞ = “Keep going forever!” (Always use parentheses with ∞ because you can never reach it!)

⚖️ Solving Linear Inequalities

Solving inequalities is almost like solving equations. You can add, subtract, multiply, and divide.

BUT THERE’S ONE MAGICAL RULE:

🔄 When you multiply or divide by a NEGATIVE number, you must FLIP THE SIGN!

Why? Let’s See!

Think about this:

• 5 > 2 ✓ (True! 5 is bigger than 2)

Now multiply both sides by -1:

• -5 > -2 ✗ (False! -5 is NOT bigger than -2!)
• -5 < -2 ✓ (True! We need to flip!)

📝 Step-by-Step Examples

Example 1: Simple Inequality

Solve: x + 4 > 10

Step 1: Subtract 4 from both sides
        x + 4 - 4 > 10 - 4
        x > 6

Answer: x > 6
Interval: (6, ∞)

Example 2: With Multiplication

Solve: 3x ≤ 15

Step 1: Divide both sides by 3
        3x ÷ 3 ≤ 15 ÷ 3
        x ≤ 5

Answer: x ≤ 5
Interval: (-∞, 5]

Example 3: The Flip Rule!

Solve: -2x > 8

Step 1: Divide by -2 (FLIP THE SIGN!)
        -2x ÷ (-2) < 8 ÷ (-2)
        x < -4

Answer: x < -4
Interval: (-∞, -4)

🔗 Compound Inequalities

A compound inequality combines two inequalities into one!

There are two types: AND and OR.

🤝 AND Inequalities

Both conditions must be true at the same time.

Think: “I need an umbrella when it’s raining AND I’m going outside.”

Example: -3 < x ≤ 5

This means:
- x is greater than -3
- AND x is 5 or less
- x is between -3 and 5

Interval: (-3, 5]

🔀 OR Inequalities

At least one condition must be true.

Think: “I’ll be happy if I get ice cream OR cake.”

Example: x < 2 OR x ≥ 7

This means:
- x is less than 2
- OR x is 7 or more

Interval: (-∞, 2) ∪ [7, ∞)

The ∪ symbol means “union” — combining two sets!

🎯 Solving Compound Inequalities

AND Example:

Solve: -4 < 2x + 2 ≤ 10

Step 1: Subtract 2 from all parts
        -4 - 2 < 2x + 2 - 2 ≤ 10 - 2
        -6 < 2x ≤ 8

Step 2: Divide all parts by 2
        -3 < x ≤ 4

Answer: -3 < x ≤ 4
Interval: (-3, 4]

OR Example:

Solve: 3x - 1 < 5 OR 2x + 3 ≥ 11

Inequality 1: 3x - 1 < 5
             3x < 6
             x < 2

Inequality 2: 2x + 3 ≥ 11
             2x ≥ 8
             x ≥ 4

Answer: x < 2 OR x ≥ 4
Interval: (-∞, 2) ∪ [4, ∞)

📈 Graphing Linear Inequalities

When we graph inequalities on a number line, we use:

• Open circle (○) for < or > (not included)
• Closed circle (●) for ≤ or ≥ (included)
• Arrow showing which direction the solutions go

Number Line Examples

x > 3
         ○━━━━━━━━━━→
    ─────┼────┼────┼────
         2    3    4

x ≤ 5
    ←━━━━━━━━━━●
    ─────┼────┼────┼────
         4    5    6

-2 < x ≤ 4
         ○━━━━━━━●
    ─────┼────┼────┼────
        -2    1    4

Graphing on a Coordinate Plane

For inequalities with two variables (like y > 2x + 1):

1. Graph the boundary line (treat it like an equation)
2. Dashed line for < or > (not included)
3. Solid line for ≤ or ≥ (included)
4. Shade the region that makes the inequality true

graph TD
    A[Graph y = 2x + 1] --> B{Is it < or >?}
    B -->|Yes| C[Draw DASHED line]
    B -->|No, ≤ or ≥| D[Draw SOLID line]
    C --> E[Test a point like 0,0]
    D --> E
    E --> F{Does point work?}
    F -->|Yes| G[Shade that side]
    F -->|No| H[Shade other side]

🧪 Example: Graph y > x + 2

1. Draw the line y = x + 2 (DASHED because >)
2. Test point (0, 0): Is 0 > 0 + 2? Is 0 > 2? NO!
3. Shade the OTHER side (above the line)

🎯 Systems of Inequalities

A system of inequalities is when you have TWO OR MORE inequalities to solve together!

The solution is the overlap — where ALL inequalities are true at once!

🏠 Real-Life Example

“I want a phone that costs less than $500 AND has at least 128GB storage.”

Both conditions must be met!

📝 Solving Systems

Example:

System:
  y > x + 1
  y ≤ -x + 5

Step 1: Graph y > x + 1
        - Dashed line through (0,1) and (1,2)
        - Shade above

Step 2: Graph y ≤ -x + 5
        - Solid line through (0,5) and (5,0)
        - Shade below

Step 3: Find the OVERLAP
        The solution is where both
        shaded regions meet!

graph TD
    A[Graph first inequality] --> B[Shade its region]
    C[Graph second inequality] --> D[Shade its region]
    B --> E[Find overlapping region]
    D --> E
    E --> F[That's your solution!]

🌟 Key Points

🎉 You Did It!

You just learned:

• ✅ What inequalities are (math statements about bigger/smaller)
• ✅ Inequality symbols (>, <, ≥, ≤, ≠)
• ✅ Interval notation (shortcuts for ranges)
• ✅ Solving linear inequalities (remember the flip rule!)
• ✅ Compound inequalities (AND means both, OR means either)
• ✅ Graphing inequalities (circles, arrows, shading)
• ✅ Systems of inequalities (find the overlap!)

🧠 The Big Idea

Inequalities aren’t about finding ONE exact answer. They’re about finding ALL the answers that work. It’s like having a VIP list instead of just one special guest!

“Life isn’t always equal. Sometimes you need more, sometimes less. Inequalities help you describe the in-between!”

🔑 Quick Reference