Problem 42

Question

Remove parentheses and simplify each expression. $$ -11 c-(4-2 c) $$

Step-by-Step Solution

Verified

Answer

The simplified expression is $-9c - 4$.

1Step 1: Remove Parentheses

To remove the parentheses in the expression $-11c - (4 - 2c)$, we first need to distribute the $-1$ which is in front of the parentheses across each term inside them. This changes the expression to:$-11c - 4 + 2c$.

2Step 2: Combine Like Terms

Now we combine the terms involving $c$. Starting with $-11c$ and $+2c$, we get: \[-11c + 2c = -9c\]Combine this with the constant term $-4$ to get the simplified expression:$-9c - 4$.

Key Concepts

Parentheses RemovalDistributive PropertyCombining Like Terms

Parentheses Removal

When working with algebraic expressions, removing parentheses is a crucial first step to simplifying them. Parentheses can be removed by distributing the sign in front of them across each term inside. This often involves negative signs. In our exercise, the expression

$-11c - (4 - 2c)$

features a negative sign in front of the parentheses. This negative sign is equivalent to multiplying each term inside the parentheses by

$-1$

So, when removing the parentheses, you apply the negative sign to both

$4$
$-2c$

This results in the expression

$-11c - 4 + 2c$

It's important to carefully change signs for each term when removing parentheses, as any mistake here can lead to incorrect results later on.

Distributive Property

The distributive property is a fundamental concept that allows us to simplify expressions by distributing a factor across terms inside parentheses. In our example, it was used implicitly to handle the negative sign. Here's how it actually works.

For an expression like $a(b + c)$, the distributive property allows you to write it as

$(ab + ac)$.

Thus, every term inside the parentheses gets multiplied by the factor outside. This property works whether the factor is positive or negative, ensuring terms inside are adjusted correctly.

In our case, the factor $-1$ is applied to each term, leading to
$-1 \times 4 = -4$
$-1 \times -2c = +2c$.

Understanding the distributive property helps simplify even the trickiest algebraic expressions effectively.

Combining Like Terms

Combining like terms is a process that helps condense algebraic expressions to their simplest form. Like terms are terms that have identical variable parts.