Evaluating Expressions with Absolute Value

TL;DR

Absolute value tells you a number's distance from zero, always resulting in a non-negative value. To evaluate expressions, first simplify anything inside the absolute value bars, then take the absolute value, and finally perform any remaining operations. It's like a special kind of parenthesis that makes its contents positive.

1. The Mental Model

Think of absolute value as a "distance meter." If you're walking on a number line, it just tells you how many steps you've taken from zero, regardless of whether you walked left (negative direction) or right (positive direction). So, 5 steps is 5, and -5 steps is also 5.

2. The Core Material

Absolute value is denoted by two vertical bars around a number or expression, like |x|. The key idea is that |x| is always positive or zero.

What Absolute Value Means

Photo by Ann H on Pexels

• |5| = 5 (5 is 5 units away from zero)
• |-5| = 5 (-5 is 5 units away from zero)
• |0| = 0 (0 is 0 units away from zero)

So, |-anything_negative| becomes positive, and |anything_positive| stays positive.

Order of Operations with Absolute Value

Photo by Ann H on Pexels

When you see absolute value in an expression, treat the absolute value bars like parentheses in the order of operations (PEMDAS/BODMAS). This means you:
1. Simplify everything inside the absolute value bars first.
2. Then, take the absolute value of that single simplified number.
3. Finally, perform any remaining operations outside the absolute value.

Let's look at an example flow:

graph TD
    A["Start with the expression"] --> B{"Is there an absolute value?"};
    B -- Yes --> C["Simplify everything INSIDE the absolute value bars"];
    C --> D["Take the absolute value (make it non-negative)"];
    D --> E["Perform any remaining operations"];
    E --> F["Get your final result"];
    B -- No --> E;

Examples of Core Evaluation:

Photo by Mikhail Nilov on Pexels

• Example 1: Evaluate |7 - 3|

  1. Simplify inside: 7 - 3 = 4
  2. Take absolute value: |4| = 4
     Result: 4

• Example 2: Evaluate |3 - 7|

  1. Simplify inside: 3 - 7 = -4
  2. Take absolute value: |-4| = 4
     Result: 4

Notice how both |7 - 3| and |3 - 7| give the same result. This is because absolute value represents distance, and the distance between 7 and 3 is the same as the distance between 3 and 7.

Expressions with Variables

Photo by Vanessa Garcia on Pexels

If there's a variable, you substitute its value first, then follow the steps.

• Example 3: Evaluate |x - 5| if x = 2

  1. Substitute x: |2 - 5|
  2. Simplify inside: | -3 |
  3. Take absolute value: 3
     Result: 3

• Example 4: Evaluate |2y + 1| if y = -4

  1. Substitute y: |2(-4) + 1|
  2. Simplify inside: |-8 + 1| = |-7|
  3. Take absolute value: 7
     Result: 7

3. Worked Example

Let's evaluate the expression: 10 - 2|3 - 5 * 2| + 7

1. Identify the absolute value part: It's |3 - 5 * 2|.
2. Simplify inside the absolute value first, following PEMDAS (multiplication before subtraction):
   • 5 * 2 = 10
   • So, |3 - 10|
   • 3 - 10 = -7
   • Now the absolute value part is |-7|.
3. Take the absolute value:
   • |-7| = 7
4. Substitute this back into the original expression:
   • The expression becomes 10 - 2 * 7 + 7. (Notice the 2 was right next to the absolute value, meaning multiplication).
5. Perform remaining operations (multiplication first, then addition/subtraction from left to right):
   • 2 * 7 = 14
   • So, 10 - 14 + 7
   • 10 - 14 = -4
   • -4 + 7 = 3

Final Result: 3

4. Key Takeaways

• Absolute value measures a number's distance from zero on a number line, always resulting in a non-negative value.
• The absolute value of a positive number is itself (|5|=5).
• The absolute value of a negative number is its positive counterpart (|-5|=5).
• Treat absolute value bars like parentheses in the order of operations: simplify inside first.
• Always complete all calculations inside the absolute value bars before taking the absolute value.
• After taking the absolute value, continue with any remaining operations in the expression.

Common Mistakes:
- Thinking |-x| = -x: No, |-5| is 5, not -5. The result of absolute value is never negative.
- Distributing into absolute value: You can't say |2 * (3-x)| = 2 * |3-x| if 3-x could be negative; always simplify inside first. Actually, |ab| = |a||b| is true, but it's safer to simplify first if you're unsure.
- Ignoring the absolute value until the very end: All operations inside the bars must be done before you take the absolute value.
- Forgetting multiplication: If you see a number right next to absolute value bars (like 2|x|), it implies multiplication.

5. Now Try It

Take 15 minutes to evaluate the following expression step-by-step. Write down each step you take, just like in the worked example.

Expression: 5 + |6 - 3 * 4| - 2^2

What success looks like:
You should arrive at a single numerical answer. Your steps should clearly show you first handled the operations inside the absolute value, then took the absolute value, then handled the exponent, and finally performed the addition and subtraction in the correct order.