Introduction

Understanding how to correctly evaluate algebraic expressions is essential for all later work in mathematics.

A reliable way to remember the order of operations is PEMDAS:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

The key idea: Always evaluate expressions in this order, even when variables are present.

Why Order Matters

Different orders of evaluation can produce different results.
For example:

• $3 + 2x$ means “multiply $2$ and $x$ first, then add $3$.”
• $(3 + 2)x$ means “add $3$ and $2$ first, then multiply by $x$.”

These are not the same expression.

Order of operations ensures that everyone interprets expressions the same way.

PEMDAS with Variables

When variables appear, PEMDAS works exactly as it does with numbers.

Parentheses

Evaluate anything inside parentheses first.

• Example: $(x + 4) - 2$
• Example: $3(2y - 1)$

Exponents

Apply exponents before multiplication or addition.

• Example: $x^2 + 3$
• Example: $(y + 1)^2$

Multiplication and Division

These occur left to right.

• Example: $4x \div 2$
• Example: $6y \cdot 3$

Addition and Subtraction

Also occur left to right.

• Example: $7 - x + 2$

Worked Examples

Example 1:

Evaluate $3 + 2x$ when $x = 5$.

• Multiply first: $2 \cdot 5 = 10$
• Then add: $3 + 10 = 13$

Example 2:

Evaluate $4(3 + y)$ when $y = 2$.

• Parentheses first: $3 + 2 = 5$
• Multiply: $4 \cdot 5 = 20$

Example 3:

Evaluate $2x^2 - 3$ when $x = 4$.

• Exponent first: $4^2 = 16$
• Multiply: $2 \cdot 16 = 32$
• Subtract: $32 - 3 = 29$

Common Mistakes to Avoid

• Forgetting parentheses:
  $3x + 2$ is not the same as $3(x + 2)$.
• Applying multiplication before exponents:
  In $2x^2$, square $x$ first, then multiply.
• Ignoring left‑to‑right rules:
  In $12 \div 3 \cdot 2$, division happens first.
• Dropping negative signs:
  $-(x + 1)$ is not the same as $-x + 1$.

Exercises

1. Evaluate $5 + 3x$ when $x = 4$.

   Solution

   $5 + 3x$ with $x = 4$ $$5 + 3(4) = 5 + 12 = 17$$
2. Evaluate $2(x + 7)$ when $x = 1$.

   Solution

   $2(x + 7)$ with $x = 1$ $$2(1 + 7) = 2 \cdot 8 = 16$$
3. Evaluate $4x^2 - 6$ when $x = 3$.

   Solution

   $4x^2 - 6$ with $x = 3$ $$4(3^2) - 6 = 4(9) - 6 = 36 - 6 = 30$$
4. Simplify $3(2y - 5)$.

   Solution

   $3(2y - 5)$
   Multiply: $3 \cdot 2y = 6y$
   Multiply: $3 \cdot (-5) = -15$
   Result: $6y - 15$
5. Evaluate $(x + 2)^2$ when $x = 5$.

   Solution

   $(x + 2)^2$ with $x = 5$ $$(5 + 2)^2 = 7^2 = 49$$
6. Simplify $12 \div 3 \cdot y$.

   Solution

   $12 \div 3 \cdot y$
   Division first: $12 \div 3 = 4$
   Then multiply: $4y$
7. Evaluate $2x - (3 - x)$ when $x = 2$.

   Solution

   $2x - (3 - x)$ with $x = 2$
   Inside parentheses: $3 - 2 = 1$
   Expression becomes $2(2) - 1 = 4 - 1 = 3$
8. Simplify $4 - 2(x - 1)$.

   Solution

   $4 - 2(x - 1)$
   Expand: $-2(x - 1) = -2x + 2$
   Combine: $4 - 2x + 2 = 6 - 2x$