Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-16, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ 4-2(8-11)+6 $$

Step-by-Step Solution

Answer

The simplified result is 16.

Step 1: Simplify Inside the Parentheses

First, solve the expression within the parentheses. In this problem, we have:\[ 8 - 11 = -3 \]

Step 2: Apply the Distributive Property

Next, substitute the result from Step 1 into the equation and apply the distributive property to remove the parentheses:\[ 4 - 2(-3) + 6 \]Multiply \(-2\) by \(-3\):\[ 4 + 6 + 6 \]

Step 3: Calculate the Result

Finally, perform the remaining addition operations to arrive at the final simplified result:\[ 4 + 6 + 6 = 16 \]

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps simplify expressions by eliminating parentheses. It allows for expansion of an expression through multiplication over addition or subtraction inside of the parentheses. In simpler terms, when you have an equation in the form of \( a(b + c) \) or \( a(b - c) \), the distributive property lets you multiply the outside term with each term inside the parentheses separately. This results in \( ab + ac \) or \( ab - ac \) respectively.

• For example, consider \( 2(x + 3) \). Applying the distributive property, you multiply \( 2 \) with \( x \) and \( 2 \) again with \( 3 \), leading to \( 2x + 6 \).
• In cases of subtraction, such as \( 3(y - 4) \), it becomes \( 3y - 12 \).

In our original problem, after simplifying inside the parentheses, we have to apply the distributive property: \( -2(-3) \) results in \( +6 \). This removes the parentheses entirely, making the expression easier to work with.

Simplifying Expressions

Simplifying expressions involves a series of steps to reduce an equation or statement into its simplest form. It often includes removing parentheses, combining like terms, and performing arithmetic operations.

• Begin by addressing any operations within parentheses first, as this will often change how the rest of the equation is handled.
• Apply properties, such as the distributive property, to deal with any multiplication or division involving terms inside parentheses.
• Combine like terms. These are terms that have the same variable raised to the same power. For example, we can combine \( 5x + 2x \) to make \( 7x \).

In our problem, simplifying \( 4 - 2(-3) + 6 \) involves first using the distributive property and then performing addition, as -2 multiplied by -3 transforms the equation to \( 4 + 6 + 6 \). The final step just involves straightforward addition.

Arithmetic Operations

Arithmetic operations are the basic manipulations we perform in mathematics and include addition, subtraction, multiplication, and division. These operations follow a specific order when performed within a complex operation, often referred to as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• **Addition (+)**: Combining two numbers to get their total sum such as \( 2 + 3 = 5 \).
• **Subtraction (-)**: Removing the value of one number from another, like \( 7 - 4 = 3 \).
• **Multiplication (×)**: Adding a number to itself a specified number of times, such as \( 4 imes 2 = 8 \).
• **Division (÷)**: Splitting a number into equal parts, for instance, \( 10 ÷ 2 = 5 \).

In the original exercise, after simplifying the expression, the final step was purely an arithmetic operation. By performing the addition \( 4 + 6 + 6 \), it provides the solution of \( 16 \). Arithmetic operations not only help in solving mathematical problems but also play a foundational role in solving algebraic expressions.