Problem 118 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify: \(3(1-2 \cdot 5)-(-28)\) (Section \(1.8,\) Example 7 )

Step-by-Step Solution

Answer

The simplified version of \(3(1-2 \cdot 5)-(-28)\) is 1.

Step 1: Performing Multiplication Inside The Parenthesis

Start the simplification by performing multiplication inside the brackets first. Which gives: \(3(1 - 2 \cdot 5) = 3(1 - 10)\)

Step 2: Perform Subtraction Inside The Parenthesis

Next, perform the subtraction operation inside the brackets: \(3(1 - 10) = 3(-9)\)

Step 3: Multiply Out The Parenthesis

Next, perform the multiplication by multiplying 3 with -9: \(3(-9) = -27\)

Step 4: Perform Subtraction with Negative Value

Finally, subtract the negative value from -27: \(-27-(-28) = -27 + 28 = 1\)

Key Concepts

Order of Operations

Understanding the order of operations is essential when simplifying algebraic expressions. This concept helps to ensure that expressions are simplified in a consistent and accurate manner.

In algebra, we follow a specific set of rules for the order in which operations should be performed. This is often remembered by the acronym PEMDAS:

• P - Parentheses first
• E - Exponents (i.e., powers and square roots, etc.)
• M & D - Multiplication and Division (from left to right)
• A & S - Addition and Subtraction (from left to right)

By following these rules, the expression is simplified correctly. For instance, in the problem \(3(1-2 \cdot 5)-(-28)\), operations within the parentheses are completed first, prioritizing multiplication over subtraction. This guarantees that everyone simplifies expressions in the same way, getting a consistent answer.

Mathematical Operations

Mathematical operations refer to the basic calculations like addition, subtraction, multiplication, and division that are used in solving expressions. Each type of operation has a specific role when it comes to simplification.

Let's explore these operations used in our problem:

• Multiplication: In the expression \(1 - 2 \cdot 5\), you start by multiplying \(2\) and \(5\), yielding \(10\).
• Subtraction: After multiplication, the expression becomes \(1 - 10\). Subtracting results in \(-9\).
• Further Multiplication: The expression \(3(-9)\) is calculated next, which gives \(-27\).
• Addition (Double-negative): Finally, you handle subtraction with negative-value \(-(-28)\), which is the same as adding \(28\) to \(-27\), resulting in \(1\).

Each operation plays a crucial role in simplifying the expression step-by-step.

Negative Numbers

Negative numbers can sometimes be confusing, especially within operations, but they follow the same basic arithmetic rules as positive numbers.

In our problem, we handle negative numbers in several steps:

• Subtracting Inside Parentheses: When the result inside the parenthesis is negative, as in \(3(1-10)\), you are fundamentally dealing with subtraction beyond zero.
• Multiplication by Negative Numbers: Remember that multiplying a positive number by a negative number, such as \(3 \times -9\), changes the sign of the product to negative \(-27\).
• Subtraction of a Negative Number (Double Negative): This operation is critical: \(-27 - (-28)\) turns into \(-27 + 28\). Adding a positive number to a negative number works similarly to subtraction but in reverse.

Understanding the simple rules governing negative numbers helps in carrying out these calculations successfully. Be sure to approach them carefully, maintain the correct operation signs, and simplify consistently.