Absolute Value: The Distance Detective 🔍

Imagine you’re a detective measuring how far things are from home base. It doesn’t matter if you walked left or right—what matters is how many steps you took!

What is Absolute Value?

Think of a number line as a big ruler laid flat on the ground. Zero is your home. Absolute value tells you how far away a number is from home—without caring which direction you went.

The Simple Rule

• Absolute value = distance from zero
• Distance is always positive (or zero)
• We write it with two vertical bars: |number|

🏠 HOME = 0

← -5  -4  -3  -2  -1   0   1   2   3   4   5 →

|3| = 3   (3 steps right from home)
|-3| = 3  (3 steps left from home)

Same distance! Both are 3 steps away.

Real-Life Example

You drop a ball. It bounces 5 feet up, then falls 5 feet down.

• Up = +5 feet
• Down = -5 feet
• The bounce height? |+5| = |-5| = 5 feet

The ball traveled the same distance each way!

graph TD
    A[Number: -7] --> B{Is it negative?}
    B -->|Yes| C[Remove the minus sign]
    B -->|No| D[Keep it as is]
    C --> E["|-7| = 7"]
    D --> F["|7| = 7"]

Quick Practice

Absolute Value Equations

Now the detective work gets exciting! Someone tells you: “I’m 5 steps from home.” Which direction did they go?

The Two-Answer Secret

When you see |x| = 5, you’re asking: “What numbers are exactly 5 steps from zero?”

Answer: TWO numbers work!

• x = 5 (5 steps right)
• x = -5 (5 steps left)

Solving Step by Step

Example: Solve |x - 2| = 7

Think: “What’s inside the bars is 7 steps from zero.”

Step 1: Set up TWO equations
   (x - 2) = 7    OR    (x - 2) = -7

Step 2: Solve each one
   x - 2 = 7           x - 2 = -7
   x = 9               x = -5

Step 3: Check both answers
   |9 - 2| = |7| = 7 ✓
   |-5 - 2| = |-7| = 7 ✓

Both x = 9 and x = -5 are correct!

graph TD
    A["|x - 2| = 7"] --> B["Split into 2 cases"]
    B --> C["x - 2 = 7"]
    B --> D["x - 2 = -7"]
    C --> E["x = 9"]
    D --> F["x = -5"]
    E --> G["✓ Both answers valid"]
    F --> G

Watch Out! Special Cases

Absolute Value Inequalities

The detective now asks: “Who lives within 4 blocks of home?” or “Who lives more than 4 blocks away?”

Less Than = Stay Close (AND)

|x| < 4 means “less than 4 steps from zero”

← -4  -3  -2  -1   0   1   2   3   4 →
     [===================]
     ↑ Everyone inside! ↑

Answer: -4 < x < 4

Memory trick: “Less than” keeps you close = use AND

Greater Than = Go Far (OR)

|x| > 4 means “more than 4 steps from zero”

← -5  -4  -3  -2  -1   0   1   2   3   4   5 →
[====]                              [====]
  ↑ Go left far                 Go right far ↑

Answer: x < -4  OR  x > 4

Memory trick: “Greater than” sends you far = use OR

Solving Example

Solve: |x + 1| ≤ 3

Step 1: This is "less than or equal" = AND case

Step 2: Split it
   -3 ≤ (x + 1) ≤ 3

Step 3: Solve for x (subtract 1 from all parts)
   -3 - 1 ≤ x ≤ 3 - 1
   -4 ≤ x ≤ 2

Answer: All numbers from -4 to 2 (including both)

graph TD
    A["Absolute Value Inequality"] --> B{"< or ≤?"}
    B -->|"Less than"| C["AND case: Stay close"]
    C --> D["-a < stuff < a"]
    B -->|"> or ≥"| E["OR case: Go far"]
    E --> F["stuff < -a OR stuff > a"]

Absolute Value Graphs

Let’s draw what absolute value looks like!

The Basic V-Shape

The graph of y = |x| makes a perfect V shape.

        y
        |
      4 |         *
        |        / \
      3 |       /   \
        |      /     \
      2 |     /       \
        |    /         \
      1 |   /           \
        |  /             \
      0 +--*---------------*-- x
       -4 -3 -2 -1  0  1  2  3  4

Why the V?

Key features:

• Vertex (tip of V) at (0, 0)
• Opens upward like a smile
• Symmetric like butterfly wings

Absolute Value Transforms

Now we can move, stretch, and flip our V!

Moving the V Around

y = |x - h| + k moves the vertex to point (h, k)

y = |x - 3| + 2
         ↑     ↑
    Right 3   Up 2

Vertex moves from (0,0) to (3, 2)

graph TD
    A["y = |x|"] --> B["Add number OUTSIDE bars"]
    B --> C["y = |x| + k → Shift UP k units"]
    A --> D["Subtract number INSIDE bars"]
    D --> E["y = |x - h| → Shift RIGHT h units"]
    A --> F["Add number INSIDE bars"]
    F --> G["y = |x + h| → Shift LEFT h units"]

Stretching and Flipping

y = a|x| changes the steepness:

Example: y = -2|x| + 5

• The -2 makes it steep AND flipped (opens down)
• The +5 moves vertex up to (0, 5)

Upside-down V:

      5 +------*------
        |     /|\
      4 |    / | \
        |   /  |  \
      3 |  /   |   \
        | /    |    \
      0 +------+------
           vertex at (0,5)

Transform Cheat Sheet

The Complete Picture

graph TD
    A["ABSOLUTE VALUE"] --> B["= Distance from Zero"]
    B --> C["Equations: |stuff| = number"]
    C --> D["2 solutions usually"]
    B --> E["Inequalities"]
    E --> F["< means AND #40;stay close#41;"]
    E --> G["> means OR #40;go far#41;"]
    B --> H["Graphs"]
    H --> I["V-shape at origin"]
    B --> J["Transforms"]
    J --> K["Move, stretch, flip the V"]

🎯 Key Takeaways

1. Absolute value = distance from zero (always positive or zero)

2. Equations give 2 answers (one positive, one negative case)

3. Inequalities:

   • Less than → AND → stay between limits
   • Greater than → OR → go beyond limits

4. Graphs: V-shape, vertex at origin

5. Transforms: Move the V with (h, k), stretch with a, flip with negative

You’re now an Absolute Value Detective! You can measure distances, solve mysteries with two answers, and draw V-shaped graphs. The unknown is no longer unknown! 🎉