Solving PEMDAS Problems with Absolute Value- A Tutorial

What the Heck Is Absolute Value, Anyway?

Absolute value sounds intimidating. It's not. It just means "how far a number is from zero" on a number line. The symbol is two vertical bars: |x|.

So |5| = 5 and |-5| = 5. Both are five units away from zero.

The negative sign inside the bars flips the number to positive. That's it. Nothing fancy.

Where Does Absolute Value Sit in PEMDAS?

PEMDAS tells you the order of operations:

Here's where people get confused: absolute value bars act like parentheses. You solve everything inside them before you drop the bars and continue.

So absolute value gets evaluated in the P step—right alongside parentheses.

The Real Rule: Inside First, Then Outside

When you see |3 - 7|, you don't just slap absolute value on the result. You solve inside first:

  1. Calculate 3 - 7 = -4
  2. Take the absolute value: |-4| = 4

That's the whole game. Inside first, then apply the absolute value.

Examples That Actually Make Sense

Basic Example

|8 - 12| + 5

Step 1: Inside the absolute value: 8 - 12 = -4

Step 2: Apply absolute value: |-4| = 4

Step 3: Add: 4 + 5 = 9

Nested Absolute Values

| |2 - 6| - 3|

Work inside out:

Step 1: Inner: 2 - 6 = -4

Step 2: Inner absolute: |-4| = 4

Step 3: Outer: |4 - 3| = |1| = 1

With Exponents

|(-2)³| + 4

Step 1: Exponent first: (-2)³ = -8

Step 2: Absolute value: |-8| = 8

Step 3: Add: 8 + 4 = 12

Comparison: How Absolute Value Stacks Up

Expression Correct Answer Common Mistake
|-8 + 2| |-6| = 6 |-8| + |2| = 10 ❌
|3 - 9| × 2 |-6| × 2 = 12 3 - |9 × 2| = -15 ❌
5 + |-4| 5 + 4 = 9 |(5 + -4)| = 1 ❌

The Mistake That Costs Everyone Points

Students love to distribute the absolute value bars like parentheses. You can't do that.

|a + b| ≠ |a| + |b|

Try it: a = 3, b = -5

|3 + (-5)| = |-2| = 2

|3| + |-5| = 3 + 5 = 8

Those aren't equal. Stop distributing. Just solve inside first.

How to Get Started: Step-by-Step

Here's your checklist for any PEMDAS problem with absolute values:

  1. Find all absolute value bars — they count as grouping symbols
  2. Solve inside each set before applying the bars
  3. Drop the bars once you've simplified the inside
  4. Continue with exponents (if any outside the bars)
  5. Handle multiplication/division
  6. Finish with addition/subtraction

Example walkthrough:

20 ÷ |4 - 8| + 3²

Step 1: Inside bars: 4 - 8 = -4

Step 2: | -4 | = 4

Step 3: Exponent: 3² = 9

Step 4: Division: 20 ÷ 4 = 5

Step 5: Add: 5 + 9 = 14

Quick Reference Table

Expression Inside First Absolute Value Final Answer
|7 - 10| 7 - 10 = -3 |-3| 3
|2 × 4| - 6 2 × 4 = 8 |8| 8 - 6 = 2
12 ÷ |1 - 4| 1 - 4 = -3 |-3| 12 ÷ 3 = 4
|(-3)²| + 1 (-3)² = 9 |9| 9 + 1 = 10

What About Negative Results?

Sometimes the final answer is negative. That's fine. The absolute value only applies to the expression inside the bars, not the whole problem.

8 - |10 - 15| = 8 - | -5 | = 8 - 5 = 3

2 - |7 - 3| = 2 - |4| = 2 - 4 = -2

Negative answers are valid. Don't force everything to be positive.

Bottom Line

Absolute value bars are grouping symbols. They get handled in the parentheses step of PEMDAS. Solve inside first, then apply the absolute value, then keep going with the rest of the problem.

Don't distribute them. Don't skip the inside step. Don't treat them like exponents or ignore them until the end.

That's all there is to it. 🔢