1.4 Properties of Real Numbers

Learning Objectives

Identify and apply the commutative property of addition and multiplication.
Identify and apply the distributive property to rewrite and simplify expressions.

Launch

In earlier sections, we followed procedures to evaluate expressions. Now that we are using variables, we need rules that tell us when expressions stay the same value. These rules are called the properties of real numbers.

Commutative Property

The commutative property says that the order of addition or multiplication does not affect the result.

Addition: $a + b = b + a$
Multiplication: $a \times b = b \times a$

📌 Note: This property does not apply to subtraction or division.

Example 1.4.1

Rewrite each using the commutative property of multiplication.

$(3)(5)$
$(-3)(9)$
$(x)(7)$
$(2y)(x)$
$(x + 1)(x + 2)$

Solution

$(3)(5) = (5)(3)$
$(-3)(9) = (9)(-3)$
$(x)(7) = (7)(x)$
$(2y)(x) = (x)(2y)$
$(x + 1)(x + 2) = (x + 2)(x + 1)$

📌 Order can change, but the product stays the same—even with binomial factors (like part e).

Example 1.4.2

Rewrite each using the commutative property of addition.

$4 + 8$
$-9 + 14$
$5 + x$
$y + x$

Solution

$4 + 8 = 8 + 4$
$-9 + 14 = 14 + (-9)$
$5 + x = x + 5$
$y + x = x + y$

📌 Changing the order does not change the sum.

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Distributive Property

The distributive property lets us multiply a single term by each term inside parentheses:

$a(b + c) = ab + ac$

It works with subtraction too:

$a(b - c) = ab - ac$

Example 1.4.3

Simplify $4(x + 5)$ .

Solution

Apply the distributive property:

$4(x + 5) = 4x + 20$

This property allows us to rewrite an expression without changing its value.

Example 1.4.4

Simplify $-5(x - 4)$ .

Solution

Distribute $-5$ to both terms:

$-5(x - 4) = -5x + 20$

Example 1.4.5

Simplify $2(4x - 3y + 5)$ .

Solution

Multiply each term by 2:

$2(4x - 3y + 5) = 8x - 6y + 10$

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Using the Distributive Property to Simplify

We often use the distributive property with negatives or when simplifying expressions.

Example 1.4.6

Simplify $-(3x + 2)$ .

Solution

Think of the negative sign as $-1$ :

$-1(3x + 2) = -3x - 2$

Example 1.4.7

Simplify $5 - (2x - 3)$ .

Solution

Distribute the negative sign:

$5 - (2x - 3) = 5 - 2x + 3 = -2x + 8$

Example 1.4.8

Simplify $3(2y - 5) + 4(y + 2)$ .

Solution

Step 1: Apply the distributive property.

$(6y - 15) + (4y + 8)$

Step 2: Combine like terms.

$6y + 4y - 15 + 8 = 10y - 7$

Example 1.4.9

Simplify $6\left(\frac{1}{6}x + \frac{2}{3}\right)$ .

Solution

Distribute the 6.

We multiply $6 \cdot \frac{1}{6}x = x$ and $6 \cdot \frac{2}{3} = 4$ .

$6\left(\frac{1}{6}x + \frac{2}{3}\right) = x + 4$

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Fill in the Blank

Sometimes we work backwards with the distributive property.

Example 1.4.10

Fill in the blank: $5(\_\_\_) = 5x + 10$

Solution

$5(x + 2) = 5x + 10$

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Essentials of Mathematics Copyright © 2026 by Scott McDaniel is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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