Problem 58 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ -3(x+1)-2 $$

Step-by-Step Solution

Answer

The simplified expression is \(-3x - 5\)

Step 1: Apply the Distributive Property

To begin with, apply the distributive property to the parenthesis. It means to multiply -3 with each term inside the parenthesis. So, -3 is multiplied with x and -3 is also multiplied with 1 giving \(-3x - 3\). Therefore, the expression becomes \(-3x - 3 - 2\)

Step 2: Combine Like Terms

The next step is to combine like terms. Like terms are terms that contain the same variables raised to the same power. In this case -3 and -2 are like terms. Combining -3 and -2 gives -5. Therefore, the simplified expression becomes \(-3x - 5\)

Key Concepts

Understanding the Distributive Property

The distributive property is a key concept in simplifying expressions. It allows you to multiply a single term across terms within parentheses. In our example, we apply it to the expression \(-3(x + 1) - 2\).

Here's how it works:

• Multiply -3 by each term inside the parentheses, which includes \(x\) and \(1\).
• This process transforms the expression into \(-3x - 3\).

By distributing, you remove the parentheses, preparing the expression for further simplification.
The distributive property can be remembered as "sharing" what's outside the parentheses with everything inside.

The Art of Combining Like Terms

Once you've applied the distributive property, the next step in simplifying is combining like terms. Like terms have the same variable raised to the same power, or they may just be constant numbers.

In the expression \(-3x - 3 - 2\):

• Notice that -3 and -2 are like terms because they are both constant numbers.
• Combining these gives \(-3 - 2 = -5\).

So, the expression simplifies to \(-3x - 5\).
Combining like terms helps to streamline the expression, making it easier to manage and understand.

Understanding Linear Expressions

A linear expression is a type of algebraic expression where the highest power of the variable is one. This means there are no squared terms or higher.

In our simplified expression \(-3x - 5\):

• \(-3x\) is a linear term because the variable \(x\) is raised to the power of one.
• -5 is a constant term.

Linear expressions are straightforward and form the foundation for solving linear equations.
Understanding them is crucial for building skills in algebra and solving more complex mathematical problems.