Problem 49 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(-(5 x+2)\)

Step-by-Step Solution

Answer

The simplified expression is \(-5x - 2\).

Step 1: Understand the Distributive Property

The distributive property states that for any numbers or variables \(a, b,\) and \(c\), the expression \(a(b+c)\) is equivalent to \(ab+ac\). In this case, we have to apply this property to distribute the negative sign through the expression \(-(5x+2)\).

Step 2: Apply the Distributive Property

Distribute the negative sign across each term inside the parentheses. This means multiplying \(-1\) by each term inside: \(-(5x+2) = -1 \times (5x) + (-1) \times 2\).

Step 3: Perform Simplification

Now, perform the multiplication. First, \(-1 \times 5x = -5x\) and \(-1 \times 2 = -2\). Thus, after simplifying the expression we get: \(-5x - 2\).

Key Concepts

Simplification

Simplification is a vital step in math that helps make expressions shorter and easier to understand. When simplifying, you need to combine like terms and perform operations to reduce an expression to its simplest form. In the case of the distributive property, simplification happens after you have distributed each term individually, as shown in our example:

• Start by multiplying each term inside the parentheses by the factor outside.
• Combine and collect like terms, if necessary.

In our original problem, after distributing the negative sign (or multiplying by \(-1\)), we end up with the terms \(-5x\) and \(-2\). Since there are no like terms to combine, the expression is already in its simplest form: \(-5x - 2\). This process helps to ensure that the expression is fully calculated and ready for use in further mathematical operations.

Negative Sign

Handling negative signs is crucial when working with expressions and equations. A negative sign in front of parentheses indicates that each term inside should be multiplied by \-1\. This can change the sign of each term to its opposite:

• If the term is positive, it becomes negative.
• If the term is negative, it becomes positive.

In our exercise, we apply the negative sign to \(5x+2\) by distributing:

• The \(5x\) becomes \(-5x\).
• The \(2\) becomes \(-2\).

This operation allows us to correctly remove the parentheses while maintaining the validity of the expression. Keeping track of negative signs helps avoid errors and ensures accurate simplification.

Multiplying Terms

Multiplication is at the core of using the distributive property, especially when dealing with multiple terms inside parentheses. Each term must be multiplied by the factor outside. In the example of \(-(5x+2)\), the negative sign acts as a \-1\ multiplier:

• Multiply \-1\ by \(5x\): \(-1 \times 5x = -5x\).
• Multiply \-1\ by \(2\): \(-1 \times 2 = -2\).

This step is essential in correctly applying the distributive property. It changes the coefficients and keeps track of negative signs, allowing you to arrive at a correctly simplified expression. Practice with multiplying terms will help solidify your understanding of these foundational algebraic skills.