Problem 50 - Prealgebra

Question

Simplify each expression. $$3-4(m+1)$$

Step-by-Step Solution

Answer

The simplified expression is \(-4m - 1\).

Step 1: Distribute the Negative Sign

We start by distributing the negative sign into the parentheses for the expression \[ 3 - 4(m+1) \]. This involves distributing the \(-4\) to both \(m\) and \(1\) inside the parentheses.This gives us:\[3 - 4m - 4 \]

Step 2: Combine Like Terms

Now, we combine the like terms in the expression \[ 3 - 4m - 4 \]. This involves combining the constants \(3\) and \(-4\).Combine them:\[ 3 - 4 = -1 \]So, the expression becomes:\[-4m - 1 \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in prealgebra. It helps to make algebraic expressions easier to manage and evaluate.
To simplify an expression, you need to follow a series of steps that include removing any parentheses, combining similar terms, and organizing the expression in its most compact form.
Consider the expression given: - Start by identifying and addressing any parentheses, just like in the problem - Use operations to break down terms where possible Simplifying might appear challenging at first, but it is simply about applying arithmetic and following the rules of algebra consistently. By tackling each element step by step, it becomes quite manageable. In our given exercise, after distributing the terms, the expression was simplified into \(-4m - 1\). It reflects fewer terms making it easier to work with in further calculations.

Distributive Property

The Distributive Property is a handy algebraic tool. Essentially, it allows you to "distribute" one term over others within parentheses. It states that \(a(b + c) = ab + ac\).
This property helps to multiply a single term by all terms inside parentheses. For example, in our problem, you start with: - \[3 - 4(m + 1)\]- Once the negative sign is distributed, \(-4\) multiplies each term inside the parentheses, producing: - \[-4m - 4\] Let's break it down: - Multiply \(-4\) with \(m\) to get \(-4m\)- Multiply \(-4\) with \(1\) to get \(-4\)After the distribution, any expression will no longer include parentheses. The role of this property is just like opening a door to solve more complex equations.

Combining Like Terms

Combining like terms is the process of simplifying expressions by merging terms that have the same variables raised to the same power.
In algebra, it's important to make equations easier to interpret by reducing the number of terms. A good example lies in our exercise: - From \[3 - 4m - 4\], identify the similar terms. In this case, the constants \(3\) and \(-4\) are like terms. - Add these constants together: \(3 + (-4) = -1\)After recognizing and then combining these similar terms, the simplified result is \(-4m - 1\). Combining like terms is one of the most useful skills in solving algebraic equations, as it tidies up expressions, leading to fewer errors and deeper insights into the solutions.