Solving $-|-6 \times -11|$: A Step-by-Step Guide

Let's break down how to simplify the expression $-|-6 ">\times -11|$. This involves understanding absolute values and basic arithmetic. Don't worry, it's simpler than it looks! We'll go through it step by step so you can easily follow along and understand the process. By the end, you'll not only know the answer but also understand the logic behind it. Let's dive in!

Understanding Absolute Value

Before we jump into the full expression, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. It's always non-negative. Think of it as stripping away the sign. So, the absolute value of 5, written as $|5|$, is 5, and the absolute value of -5, written as $|-5|$, is also 5. Basically, it's the magnitude without the direction. This concept is crucial for solving our problem. Understanding absolute values helps in various mathematical contexts, especially when dealing with distances or magnitudes where the direction isn't important. For example, in physics, when calculating the magnitude of a vector, you're essentially using the concept of absolute value in a multi-dimensional space. So, remember, absolute value always gives you a non-negative result. It’s a fundamental concept that pops up in many different areas of math and science.

Step-by-Step Simplification

Now, let's tackle the expression $-|-6 ">\times -11|$ step by step:

1. Multiplication Inside the Absolute Value: First, we need to deal with the multiplication inside the absolute value symbols: $-6 ">\times -11$. When you multiply two negative numbers, you get a positive number. So, $-6 ">\times -11 = 66$. This simplifies our expression to $-|66|$.
2. Absolute Value: Next, we find the absolute value of 66. Since 66 is already a positive number, its absolute value is just 66. Therefore, $|66| = 66$. Now our expression looks like $-66$.
3. Final Step: Finally, we apply the negative sign that's sitting outside the absolute value. So, we have $-(66)$, which is simply $-66$.

Therefore, $-|-6 ">\times -11| = -66$.

Detailed Explanation

Let's delve deeper into each step to ensure you've got a solid grasp. We start with $-|-6 ">\times -11|$. The key here is to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our case, we focus on what's inside the absolute value first.

Inside the absolute value, we have a multiplication: $-6 ">\times -11$. A negative number multiplied by another negative number results in a positive number. Specifically, $-6 ">\times -11 = 66$. So the expression inside the absolute value simplifies to 66. Now we have $-|66|$.

The next step is to evaluate the absolute value. The absolute value of a number is its distance from zero. For positive numbers, the absolute value is the number itself. So, $|66| = 66$. Our expression now looks like -66.

Finally, we bring down the negative sign that was outside the absolute value. This means we take the negative of the absolute value we just calculated. Therefore, $-(66) = -66$. This is our final answer. It's crucial to remember that the absolute value always returns a non-negative number, but the negative sign outside can make the final result negative.

Common Mistakes to Avoid

When simplifying expressions like these, there are a few common pitfalls you might encounter. Let's highlight some of them to help you avoid making these mistakes:

• Forgetting the Order of Operations: Always remember PEMDAS. Make sure you perform the multiplication inside the absolute value before dealing with the absolute value itself. Skipping this can lead to incorrect results.
• Misunderstanding Absolute Value: The absolute value always returns a non-negative number. Don't forget to apply this rule. For example, $|-5|$ is 5, not -5.
• Ignoring the Outer Negative Sign: The negative sign outside the absolute value is easily overlooked. Remember to apply it after you've calculated the absolute value.
• Incorrect Multiplication of Negative Numbers: A negative number multiplied by a negative number is always positive. Forgetting this basic rule will throw off your entire calculation.

By keeping these common mistakes in mind, you can increase your accuracy and confidence when tackling similar problems.

Practice Problems

To solidify your understanding, let's try a few practice problems:

1. Simplify $-|-4 ">\times -5|$
2. Simplify $-|2 ">\times -7|$
3. Simplify $-|-3 ">\times 8|$

Try solving these on your own, and then check your answers. Remember to follow the steps we discussed earlier:

• Multiply inside the absolute value.
• Find the absolute value.
• Apply the outer negative sign.

Working through these practice problems will help reinforce the concepts and build your problem-solving skills. Don't just memorize the steps; understand why each step is necessary.

Solutions to Practice Problems

Here are the solutions to the practice problems:

1. $-|-4 ">\times -5| = -|20| = -20$
2. $-|2 ">\times -7| = -|-14| = -14$
3. $-|-3 ">\times 8| = -|-24| = -24$

Check your work and see if you arrived at the same answers. If not, review the steps and try to identify where you went wrong. Understanding your mistakes is a crucial part of the learning process.

Real-World Applications

While it might seem abstract, absolute value has many real-world applications. It's used in various fields like physics, engineering, and computer science.

• Physics: In physics, absolute value is used to calculate the magnitude of vectors, which represent quantities with both magnitude and direction. For example, when calculating the speed of an object (which is the magnitude of its velocity), you're using absolute value.
• Engineering: Engineers use absolute value to calculate tolerances in manufacturing. Tolerance is the allowable variation in a dimension. Using absolute value ensures that the variation is always a positive value, regardless of whether the actual dimension is larger or smaller than the specified dimension.
• Computer Science: In computer science, absolute value is used in various algorithms, such as those involving distance calculations or error measurements. For instance, when calculating the difference between two values, absolute value ensures that the result is always a positive quantity, representing the magnitude of the difference.

By understanding these real-world applications, you can see that absolute value is not just a theoretical concept but a practical tool used in many different fields.

The correct answer is C. -66. Hope this helps you guys!