Problem 46 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$-(-3 x+1)$$

Step-by-Step Solution

Answer

The expanded expression is \(3x - 1\).

Step 1: Identify the Expression

We need to distribute a negative sign across the expression inside the parentheses, which is \(-(-3x + 1)\). The negative sign outside is the same as multiplying by \(-1\).

Step 2: Distribute the Negative Sign

To apply the distributive property, the negative sign (or \(-1\)) is multiplied by each term inside the parentheses. This results in two individual multiplications: \(-1\times -3x\) and \(-1\times 1\).

Step 3: Perform the Multiplications

First, multiply \(-1\times -3x\) which results in \(3x\). Then multiply \(-1\times 1\) which results in \(-1\).

Step 4: Write the Expanded Expression

Combine the results from Step 3 to obtain the distributed expression: \(3x - 1\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (like addition, subtraction, and multiplication). It represents a mathematical phrase that can be simplified or evaluated for different values of the variables included. In our exercise, \(-(-3x + 1)\) is an example of an algebraic expression. It includes a variable term, \(-3x\), and a constant, 1, both contained within parentheses and preceded by a negative sign.

In algebraic expressions like this one, terms can be arranged and manipulated using various properties of arithmetic, like the distributive property. This allows us to simplify or modify expressions based on their structure, making them easier to work with when solving equations or evaluating situations.

Multiplication

Multiplication is one of the four basic arithmetic operations and is essentially repeated addition. When working with algebraic expressions, multiplication allows us to scale terms by constants or other variables.
For instance, when using the distributive property in our example, each term inside the parentheses \((-3x + 1)\) is multiplied by \(-1\). This highlights a fundamental property of multiplication in algebra: distributing a multiplier across an expression inside parentheses.

• When multiplying two numbers (or a number and a variable), simply multiply them directly.
• When multiplying variables, adhere to the rules of exponents if necessary.

In our solution, the multiplication was simple: \(-1 \times -3x = 3x\) and \(-1 \times 1 = -1\). Understanding multiplication's role in this process clarifies how expressions are expanded and simplified.

Negative Numbers

Negative numbers are values less than zero, symbolized by a minus sign. They play a critical role in algebra, and their properties must be understood to correctly perform operations like multiplication or addition with them. In our task, we encounter negative numbers in two key areas:

1. **Multiplication involving negatives:** Understanding that multiplying a negative by a negative results in a positive is crucial. For example, \(-1 \times -3x\) results in \(+3x\). This result is because the two negatives "cancel" each other out.

2. **Distributing a negative sign:** The negative outside the brackets changes the sign of each term inside when distributed. This follows from the first point, where multiplication by \(-1\) changes the sign of a term.

• A negative times a positive (or vice versa) yields a negative.
• A negative times a negative yields a positive.

Recognizing and applying these properties is crucial for tackling a wide array of algebra problems effectively.