Problem 105 - Intermediate Algebra

Question

Perform each indicated operation. See Sections 1.4 and 5.4 $$ -3(x+5) $$

Step-by-Step Solution

Answer

The simplified expression is \(-3x - 15\).

Step 1: Distribute the Negative Sign

In the expression \(-3(x+5)\), the negative sign in front of the 3 can be distributed across the terms inside the parentheses. This means you will multiply both \(x\) and 5 by \(-3\).

Step 2: Multiply Each Term By \(-3\)

First, multiply \(-3\) by \(x\). This gives \(-3x\). Next, multiply \(-3\) by 5, which results in \(-15\).

Step 3: Write the Simplified Expression

Combine the products from the previous step into a single expression: \(-3x - 15\). This is the simplified form of the original expression \(-3(x+5)\).

Key Concepts

Negative Sign Distribution

The concept of negative sign distribution is crucial in understanding how to effectively simplify expressions, especially when dealing with a negative number outside of parentheses. The negative sign in front of a number or variable indicates that you are multiplying by \(-1\). In the expression \(-3(x+5)\), the negative sign along with the number \(-3\) needs to be applied to each term inside the parentheses.

• This process involves taking the coefficient in front, which is \(-3\), and distributing it to each term inside the parentheses. In our case, the terms are \(x\) and \(+5\).
• When you multiply \(-3\) by \(x\), you get \(-3x\).
• Then, \(-3\) times \(+5\) results in \(-15\). Notice how the negative sign changes the sign of the term from positive to negative.

Understanding and applying this correctly ensures that expressions are simplified accurately.

Simplifying Expressions

Simplifying expressions in algebra involves combining like terms and reducing expressions to their simplest form. Once you have distributed all terms outside the parentheses, as shown in the previous step, the next task is to combine the resulting terms into a single expression.

• In the expression \(-3(x+5)\), once we distribute, we get \(-3x\) and \(-15\).
• These terms are then written together as a single simplified expression: \(-3x - 15\).

There are no like terms to combine further in our example, but in some cases, you may need to look for and combine like terms to fully simplify.
It's essential to handle negative and positive signs correctly during simplification to avoid errors.

Algebraic Operations

Algebraic operations encompass actions such as addition, subtraction, multiplication, and division applied to algebraic expressions. These operations follow specific mathematical rules and properties, such as the distributive property, which we utilized to solve the problem at hand.

• The distributive property allows us to remove parentheses by distributing the multiplication over addition or subtraction within them, as shown with the term \(-3(x+5)\).
• Each term within the parentheses is dealt with separately, ensuring the final expression is correctly formed.
• It's important to apply these operations systematically to ensure consistency and accuracy in solving algebraic problems.

Mastering these operations, particularly the application of the distributive property, is key to solving more complex algebraic equations efficiently. This foundational understanding facilitates the progression to higher levels of algebra.