Problem 54 - Prealgebra

Question

Perform the indicated operations. $$-6(5 x+4)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-30x - 24\).

Step 1: Distribute the Negative Sign

To solve the expression \(-6(5x+4)\), we need to distribute the \(-6\) to both terms inside the parentheses. This means multiplying \(-6\) by \(5x\) and by \(4\).

Step 2: Multiply \(-6\) by \(5x\)

First, multiply \(-6\) by \(5x\):\[-6 \times 5x = -30x\].

Step 3: Multiply \(-6\) by \(4\)

Next, multiply \(-6\) by \(4\):\[-6 \times 4 = -24\].

Step 4: Combine the Products

Combine the results from Step 2 and Step 3 into a single expression:\[-30x - 24\].

Key Concepts

Multiplying Integers

Multiplying integers involves taking two whole numbers  which can be either positive or negative  and finding their product. When you multiply two integers, different rules apply depending on the sign of each number. If you're multiplying two positive integers, the result is positive. Similarly, two negative integers multiply to give a positive product.
However, if you multiply an integer with a different sign, one positive and one negative, the product will be negative:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Negative × Positive = Negative
• Positive × Negative = Negative

Understanding these rules helps when distributing multiplication over addition or subtraction in algebraic expressions. It ensures that the signs are correct in each step of the calculation.

Combining Like Terms

Combining like terms is a fundamental algebraic technique used to simplify expressions or equations. "Like terms" are terms that have the same variable raised to the same power. For example, in the expression \(-30x - 24\), the term \(-30x\) is made up of "like terms" bereft of its equivalent.
To combine like terms:

• Identify terms with identical variable parts.
• Add or subtract their coefficients as indicated by the operation signs.
• Carry forward any constant terms separately.

In our expression -30x - 24, no further simplification is possible since -30x is a variable term and -24 is a constant. Thus, combining like terms aids in reducing expressions to simpler forms, making it easier to solve or visualize.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operational symbols such as plus "+" or minus "-" signs. They represent a value and are used to model real-world situations or solve problems.
An algebraic expression consists mostly of:

• Variables: Letters that stand for unknown numbers, like \(x\) in our example.
• Constants: Fixed numbers, in this case, die 2424 which do not change their value.
• Coefficients: The numerical part multiplied by the variable, such as the \(-30\) in \(-30x\).

The key to working with algebraic expressions is understanding how to manipulate these parts, particularly how to apply operations like multiplication (using the distributive property) and rearrangement (using like terms). As seen in our exercise, correctly distributing and combining terms brings an expression to its simplest form.