Distributive Property: Simplify -2(12x - 1) Easily

In mathematics, simplifying expressions is a fundamental skill. One of the most common techniques used to achieve this is the distributive property. This property allows us to multiply a single term by two or more terms inside a set of parentheses. When dealing with a negative number outside the parentheses, it's crucial to apply the distributive property carefully to ensure the correct signs for each term. In this article, we'll break down the process of simplifying the expression $-2(12x - 1)$ step by step, making it easy to understand and apply to similar problems. This method is not only useful for basic algebra but also forms the foundation for more complex mathematical operations. Understanding how to accurately distribute a negative number is essential for solving equations, simplifying polynomials, and tackling various algebraic challenges. Let's dive in and demystify the process!

Understanding the Distributive Property

The distributive property states that for any numbers a, b, and c, the following is true:

• a( b + c ) = a b + a c

This property extends to subtraction as well:

• a( b - c ) = a b - a c

The distributive property is a cornerstone of algebra. It allows us to handle expressions that involve parentheses by multiplying the term outside the parentheses with each term inside. This is particularly useful when simplifying expressions and solving equations. The core idea is to distribute the multiplication across the addition or subtraction within the parentheses, ensuring that each term inside is properly accounted for. When used correctly, the distributive property transforms a complex expression into a more manageable form, making it easier to combine like terms and arrive at a simplified result. This property is not just a mathematical trick; it’s a fundamental concept that underpins many algebraic manipulations, and a solid understanding of it is crucial for success in mathematics.

Step-by-Step Simplification of $-2(12x - 1)$

Let's simplify the expression $-2(12x - 1)$ using the distributive property.

Step 1: Apply the Distributive Property

We need to distribute $-2$ to both terms inside the parentheses:

• $-2 * (12x) + (-2) * (-1)$

Step 2: Perform the Multiplication

Now, multiply $-2$ by each term:

• $-2 * 12x = -24x$
• $-2 * -1 = 2$

Step 3: Combine the Results

Combine the results from the multiplication:

• $-24x + 2$

So, the simplified expression is $-24x + 2$.

Detailed Explanation of Each Step

To ensure a thorough understanding, let's delve deeper into each step of the simplification process. This will help clarify any potential confusion and reinforce the application of the distributive property. Understanding each step in detail can improve your problem-solving skills and ensure accurate results when working with algebraic expressions.

Step 1: Applying the Distributive Property with a Negative Number

When applying the distributive property with a negative number, it's essential to pay close attention to the signs. The expression $-2(12x - 1)$ requires us to multiply $-2$ by both $12x$ and $-1$. This is where many students can make mistakes, especially with the negative signs. Remember that multiplying a negative number by a positive number results in a negative number, and multiplying a negative number by a negative number results in a positive number. Write out each multiplication explicitly to avoid errors:

• $-2 * (12x)$
• $-2 * (-1)$

Step 2: Performing the Multiplication Correctly

Next, perform each multiplication carefully. When multiplying $-2$ by $12x$, we multiply the coefficients (the numerical parts) and keep the variable $x$:

• $-2 * 12x = -24x$

When multiplying $-2$ by $-1$, remember that two negatives make a positive:

• $-2 * -1 = 2$

Step 3: Combining the Results Accurately

Finally, combine the results to form the simplified expression. The result of the first multiplication is $-24x$, and the result of the second multiplication is $2$. Therefore, the simplified expression is:

• $-24x + 2$

There are no like terms to combine, so this is our final simplified expression. Always double-check your work to ensure that you have correctly applied the distributive property and handled the signs accurately. This careful approach will minimize errors and build confidence in your algebraic skills.

Common Mistakes to Avoid

When applying the distributive property, especially with negative numbers, it's easy to make mistakes. Here are some common errors to watch out for:

• Sign Errors: Forgetting to apply the negative sign correctly. Remember that a negative times a negative is a positive.
• Incorrect Multiplication: Multiplying the coefficients incorrectly.
• Forgetting to Distribute: Failing to distribute the term to all terms inside the parentheses.

To avoid these mistakes, double-check each step and pay close attention to the signs. Writing out each step explicitly can also help minimize errors. Practice and careful attention to detail are key to mastering the distributive property.

Practice Problems

To solidify your understanding, try simplifying these expressions using the distributive property:

1. $-3(4x + 2)$
2. $5(2x - 3)$
3. $-1(7x - 5)$

Solutions

1. $-3(4x + 2) = -12x - 6$
2. $5(2x - 3) = 10x - 15$
3. $-1(7x - 5) = -7x + 5$

Real-World Applications

The distributive property is not just a theoretical concept; it has numerous real-world applications. For example, consider calculating the total cost of purchasing multiple items at a store. If you buy 3 items that each cost 10, you can calculate the total cost by multiplying 3 by (x + 10). The distributive property is also used extensively in engineering, physics, and computer science to simplify complex calculations and model real-world phenomena. From designing structures to analyzing data, the distributive property plays a crucial role in solving practical problems.

Conclusion

Simplifying expressions using the distributive property is a fundamental skill in algebra. By carefully applying the distributive property and paying attention to signs, you can accurately simplify expressions like $-2(12x - 1)$. Remember to take your time, double-check your work, and practice regularly to build confidence and proficiency. With a solid understanding of the distributive property, you'll be well-equipped to tackle more complex mathematical problems. For further learning and practice, you might find helpful resources on websites like Khan Academy.