Problem 50 - Algebra for College Students

Question

Perform the indicated operations. $$(6 x+4)-(4 x-2)-(-x-1)$$

Step-by-Step Solution

Answer

The simplified expression is \(3x + 7\).

Step 1: Distribute Negative Signs

First, we need to distribute the negative signs in the expression \((6x + 4) - (4x - 2) - (-x - 1)\). This gives us: \(6x + 4 - 4x + 2 + x + 1\).

Step 2: Combine Like Terms - x Terms

Identify and combine the like terms involving \(x\). In the expression \(6x + 4 - 4x + 2 + x + 1\), the \(x\) terms are \(6x - 4x + x\). Combine these to get \((6 - 4 + 1)x = 3x\).

Step 3: Combine Like Terms - Constant Terms

Next, combine the constant terms in the expression.The constant terms are \(4 + 2 + 1\). Combine these to get \(4 + 2 + 1 = 7\).

Step 4: Write the Simplified Expression

Now, combine the results from Step 2 and Step 3 to write the simplified expression.This gives us \(3x + 7\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a fundamental process that simplifies an expression. It involves grouping together terms that have the same variable raised to the same power. For instance, in the expression \(6x - 4x + x\), all terms have the variable \(x\), so we can combine them easily. Here’s how you do it:

• Look for terms that share the same variable. For example, all the terms with \(x\) are like terms.
• Add or subtract the coefficients (numerical values in front of the variables). In our example, the coefficients are 6, -4, and 1.

This leads us to the combined expression \( (6 - 4 + 1)x = 3x \). Combining like terms reduces complexity and makes equations easier to solve. It's like consolidating your grocery list by grouping similar items together.

Distributive Property

The distributive property is an essential tool for managing expressions involving parentheses. It allows you to distribute a factor across a sum or difference inside parentheses. For example, if you have an expression like \((a - b) - (-c - d)\), applying the distributive property involves the following steps:

• "Distribute" the minus sign across each term inside the parentheses. This means changing the sign of each term involved. In the expression \((6x + 4) - (4x - 2) - (-x - 1)\), it becomes \(6x + 4 - 4x + 2 + x + 1\).
• Remember that subtracting a negative is the same as adding a positive: \(-(-x)\) becomes \(+x\) and \(-(-1)\) becomes \(+1\).

Applying the distributive property simplifies expressions by removing parentheses, setting the stage for further simplifications.

Simplifying Expressions

Simplifying expressions in algebra involves using several techniques to make an expression easier to work with. The overall goal is to express an equation or expression in the simplest form possible:

• First, begin by using the distributive property to eliminate parentheses, as demonstrated in the previous section.
• Next, combine like terms by adding or subtracting coefficients of terms that have the same variable.
• After combining like terms, you'll arrive at a neatly packaged expression, like \(3x + 7\).

This final expression is much easier to handle, whether you're solving for \(x\), further simplifying, or substituting values. Simplifying expressions reduces clutter and makes algebraic equations and expressions less daunting to tackle. It’s much like cutting through tangled ropes to find a straight path!