Learn on PengiOpenstax Prealgebre 2EChapter 7: The Properties of Real Numbers

Lesson 3: Distributive Property

In this prealgebra lesson from OpenStax Prealgebra 2E, students learn to simplify and evaluate algebraic expressions using the Distributive Property, applying the rule a(b + c) = ab + ac to expand terms such as 3(x + 4) into 3x + 12. Practice problems progress from basic distribution to multi-term expressions involving decimals, fractions, and negative numbers. Real-world word problems, such as calculating stamp costs and counting cash, reinforce how the property works in everyday contexts.

Section 1

📘 Distributive Property

New Concept

The Distributive Property is a key tool for simplifying algebraic expressions. It allows you to multiply a single term by each term inside parentheses, which is essential for removing them and combining like terms.

What’s next

Now you're ready to master this property. You’ll work through interactive examples, practice cards, and challenge problems to build your skills.

Section 2

Distributive Property

Property

If $a$ , $b$ , $c$ are real numbers, then

a(b + c) = ab + ac

Other forms of the distributive property include:

a(b - c) = ab - ac

(b + c)a = ba + ca

Examples

To simplify $3(x + 4)$ , distribute the $3$ to each term inside: $3 \cdot x + 3 \cdot 4$ , which results in $3x + 12$ .

To simplify $m(n - 4)$ , distribute the $m$ : $m \cdot n - m \cdot 4$ , which results in $mn - 4m$ .

Section 3

Distributing fractions and decimals

Property

The Distributive Property applies to all real numbers, including fractions and decimals. The process is the same: multiply the outside term by each term inside the parentheses and simplify the resulting products.

Examples

To simplify $\frac{3}{4}(n + 12)$ , distribute $\frac{3}{4}$ : $\frac{3}{4} \cdot n + \frac{3}{4} \cdot 12$ , which simplifies to $\frac{3}{4}n + 9$ .

To simplify $8(\frac{3}{8}x + \frac{1}{4})$ , distribute the $8$ : $8 \cdot \frac{3}{8}x + 8 \cdot \frac{1}{4}$ , which simplifies to $3x + 2$ .

Section 4

Distributing a negative number

Property

When you distribute a negative number, you must multiply the negative number by each term inside the parentheses. Be careful to get the signs correct. Remember that $-a$ is equivalent to $-1 \cdot a$ .

Examples

To simplify $-2(4y + 1)$ , distribute the $-2$ : $(-2) \cdot 4y + (-2) \cdot 1$ , which results in $-8y - 2$ .

To simplify $-11(4 - 3a)$ , distribute the $-11$ : $(-11) \cdot 4 - (-11) \cdot 3a$ , which results in $-44 + 33a$ .

Section 5

Distributing within an expression

Property

When using the order of operations, if an expression inside parentheses cannot be simplified, use the Distributive Property to remove the parentheses. After distributing, combine any like terms to finish simplifying.

Examples

To simplify $8 - 2(x + 3)$ , first distribute the $-2$ to get $8 - 2x - 6$ . Then, combine like terms ( $8$ and $-6$ ) to get $-2x + 2$ .

To simplify $4(x - 8) - (x + 3)$ , distribute both terms: $4x - 32 - x - 3$ . Combine like terms ( $4x, -x$ and $-32, -3$ ) to get $3x - 35$ .

Book overview

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Chapter 7: The Properties of Real Numbers

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Lesson 1
Lesson 1: Rational and Irrational Numbers
Learn Practice
Lesson 2
Lesson 2: Commutative and Associative Properties
Learn Practice
Lesson 3Current
Lesson 3: Distributive Property
Learn Practice
Lesson 4
Lesson 4: Properties of Identity, Inverses, and Zero
Learn Practice
Lesson 5
Lesson 5: Systems of Measurement
Learn Practice

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

📘 Distributive Property

New Concept

What’s next

Now you're ready to master this property. You’ll work through interactive examples, practice cards, and challenge problems to build your skills.

Section 2

Distributive Property

Property

If $a$ , $b$ , $c$ are real numbers, then

a(b + c) = ab + ac

Other forms of the distributive property include:

a(b - c) = ab - ac

(b + c)a = ba + ca

Examples

To simplify $3(x + 4)$ , distribute the $3$ to each term inside: $3 \cdot x + 3 \cdot 4$ , which results in $3x + 12$ .

To simplify $m(n - 4)$ , distribute the $m$ : $m \cdot n - m \cdot 4$ , which results in $mn - 4m$ .

Section 3

Distributing fractions and decimals

Property

Examples

To simplify $\frac{3}{4}(n + 12)$ , distribute $\frac{3}{4}$ : $\frac{3}{4} \cdot n + \frac{3}{4} \cdot 12$ , which simplifies to $\frac{3}{4}n + 9$ .

To simplify $8(\frac{3}{8}x + \frac{1}{4})$ , distribute the $8$ : $8 \cdot \frac{3}{8}x + 8 \cdot \frac{1}{4}$ , which simplifies to $3x + 2$ .

Section 4

Distributing a negative number

Property

Examples

To simplify $-2(4y + 1)$ , distribute the $-2$ : $(-2) \cdot 4y + (-2) \cdot 1$ , which results in $-8y - 2$ .

To simplify $-11(4 - 3a)$ , distribute the $-11$ : $(-11) \cdot 4 - (-11) \cdot 3a$ , which results in $-44 + 33a$ .

Section 5

Distributing within an expression

Property

Examples

To simplify $8 - 2(x + 3)$ , first distribute the $-2$ to get $8 - 2x - 6$ . Then, combine like terms ( $8$ and $-6$ ) to get $-2x + 2$ .

To simplify $4(x - 8) - (x + 3)$ , distribute both terms: $4x - 32 - x - 3$ . Combine like terms ( $4x, -x$ and $-32, -3$ ) to get $3x - 35$ .

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 7: The Properties of Real Numbers

Back to Openstax Prealgebre 2E

Lesson 1
Lesson 1: Rational and Irrational Numbers
Learn Practice
Lesson 2
Lesson 2: Commutative and Associative Properties
Learn Practice
Lesson 3Current
Lesson 3: Distributive Property
Learn Practice
Lesson 4
Lesson 4: Properties of Identity, Inverses, and Zero
Learn Practice
Lesson 5
Lesson 5: Systems of Measurement
Learn Practice

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

📘 Distributive Property

New Concept

What’s next

Now you're ready to master this property. You’ll work through interactive examples, practice cards, and challenge problems to build your skills.

Section 2

Distributive Property

Property

If $a$ , $b$ , $c$ are real numbers, then

a(b + c) = ab + ac

Other forms of the distributive property include:

a(b - c) = ab - ac

(b + c)a = ba + ca

Examples

To simplify $3(x + 4)$ , distribute the $3$ to each term inside: $3 \cdot x + 3 \cdot 4$ , which results in $3x + 12$ .

To simplify $m(n - 4)$ , distribute the $m$ : $m \cdot n - m \cdot 4$ , which results in $mn - 4m$ .

Section 3

Distributing fractions and decimals

Property

Examples

To simplify $\frac{3}{4}(n + 12)$ , distribute $\frac{3}{4}$ : $\frac{3}{4} \cdot n + \frac{3}{4} \cdot 12$ , which simplifies to $\frac{3}{4}n + 9$ .

To simplify $8(\frac{3}{8}x + \frac{1}{4})$ , distribute the $8$ : $8 \cdot \frac{3}{8}x + 8 \cdot \frac{1}{4}$ , which simplifies to $3x + 2$ .

Section 4

Distributing a negative number

Property

Examples

To simplify $-2(4y + 1)$ , distribute the $-2$ : $(-2) \cdot 4y + (-2) \cdot 1$ , which results in $-8y - 2$ .

To simplify $-11(4 - 3a)$ , distribute the $-11$ : $(-11) \cdot 4 - (-11) \cdot 3a$ , which results in $-44 + 33a$ .

Section 5

Distributing within an expression

Property

Examples

To simplify $8 - 2(x + 3)$ , first distribute the $-2$ to get $8 - 2x - 6$ . Then, combine like terms ( $8$ and $-6$ ) to get $-2x + 2$ .

To simplify $4(x - 8) - (x + 3)$ , distribute both terms: $4x - 32 - x - 3$ . Combine like terms ( $4x, -x$ and $-32, -3$ ) to get $3x - 35$ .

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 7: The Properties of Real Numbers

Back to Openstax Prealgebre 2E

Lesson 1
Lesson 1: Rational and Irrational Numbers
Learn Practice
Lesson 2
Lesson 2: Commutative and Associative Properties
Learn Practice
Lesson 3Current
Lesson 3: Distributive Property
Learn Practice
Lesson 4
Lesson 4: Properties of Identity, Inverses, and Zero
Learn Practice
Lesson 5
Lesson 5: Systems of Measurement
Learn Practice

Lesson 3: Distributive Property

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 7: The Properties of Real Numbers

Lesson 1: Rational and Irrational Numbers

Lesson 2: Commutative and Associative Properties

Lesson 3: Distributive Property

Lesson 4: Properties of Identity, Inverses, and Zero

Lesson 5: Systems of Measurement

Lesson overview

New Concept

What’s next

Property

Examples

Property

Examples

Property

Examples

Property

Examples

Chapter 7: The Properties of Real Numbers

Lesson 1: Rational and Irrational Numbers

Lesson 2: Commutative and Associative Properties

Lesson 3: Distributive Property

Lesson 4: Properties of Identity, Inverses, and Zero

Lesson 5: Systems of Measurement

Lesson 1: Rational and Irrational Numbers

Lesson 2: Commutative and Associative Properties

Lesson 3: Distributive Property

Lesson 4: Properties of Identity, Inverses, and Zero

Lesson 5: Systems of Measurement