Simplify The Expression Absolute Value Problem A Step-by-Step Guide

by ADMIN 68 views

In the realm of mathematics, expressions involving absolute values often present a unique challenge. The absolute value of a number is its distance from zero, irrespective of its sign. This inherent property can lead to seemingly complex expressions that require careful simplification. In this comprehensive guide, we will delve into the intricacies of simplifying the expression ∣x−7∣−∣7−x∣−∣2x−10∣∣5−x∣{ \frac{|x-7| - |7-x| - |2x-10|}{|5-x|} } and arrive at the correct solution. We will explore the fundamental concepts of absolute values, discuss the properties that govern their behavior, and apply these principles to systematically simplify the given expression. Our journey will not only provide a solution to this specific problem but also equip you with the skills and knowledge to tackle similar challenges involving absolute values with confidence. Understanding absolute value simplification is not just an academic exercise; it has practical applications in various fields, including physics, engineering, and computer science. By mastering the techniques presented here, you will enhance your problem-solving abilities and gain a deeper appreciation for the elegance and power of mathematical tools.

Understanding Absolute Values

To effectively simplify expressions involving absolute values, it is crucial to grasp the fundamental concept of absolute value itself. The absolute value of a number, denoted by |x|, represents its distance from zero on the number line. This distance is always non-negative, meaning it is either positive or zero. Mathematically, we define the absolute value of x as follows:

|x| = x, if x ≥ 0

|x| = -x, if x < 0

This definition implies that if x is a positive number or zero, its absolute value is simply x itself. However, if x is a negative number, its absolute value is the negation of x, which makes it positive. For example, |5| = 5, and |-5| = -(-5) = 5. The absolute value function essentially strips away the sign of a number, leaving only its magnitude. This seemingly simple concept has profound implications when dealing with expressions that involve absolute values.

When simplifying such expressions, we often need to consider different cases based on the sign of the expression inside the absolute value. This is because the absolute value function behaves differently depending on whether its argument is positive or negative. For instance, in the given expression, we have |x-7|, |7-x|, |2x-10|, and |5-x|. To simplify these terms, we need to determine the intervals of x for which each expression inside the absolute value is positive or negative. This involves solving inequalities and carefully considering the critical points where the expressions change sign. Mastering the concept of absolute value is the cornerstone of simplifying expressions like the one we are tackling. It lays the foundation for understanding how to handle different cases and apply the appropriate rules to arrive at the correct solution.

Properties of Absolute Values

Before we delve into the simplification process, let's explore some key properties of absolute values that will prove invaluable in our task. These properties provide us with tools to manipulate expressions involving absolute values and make them easier to handle. Here are some of the most important properties:

  1. |a| = |-a|: This property states that the absolute value of a number is equal to the absolute value of its negative. For example, |3| = |-3| = 3. This property is a direct consequence of the definition of absolute value as the distance from zero.
  2. |ab| = |a||b|: The absolute value of a product is equal to the product of the absolute values. For instance, |(2)(-3)| = |-6| = 6, and |2||-3| = 2 * 3 = 6. This property allows us to separate the absolute value of a product into the product of individual absolute values.
  3. |a/b| = |a|/|b| (where b ≠ 0): The absolute value of a quotient is equal to the quotient of the absolute values, provided the denominator is not zero. For example, |6/-2| = |-3| = 3, and |6|/|-2| = 6/2 = 3. This property is analogous to the product property and extends the concept to division.
  4. |a - b| = |b - a|: The absolute value of the difference between two numbers is the same regardless of the order of subtraction. For instance, |5 - 2| = |3| = 3, and |2 - 5| = |-3| = 3. This property is particularly useful when dealing with expressions like |x - 7| and |7 - x|, as we will see in our problem.
  5. |a + b| ≤ |a| + |b|: This is the triangle inequality, which states that the absolute value of a sum is less than or equal to the sum of the absolute values. For example, |2 + (-3)| = |-1| = 1, and |2| + |-3| = 2 + 3 = 5. The triangle inequality has broad applications in various areas of mathematics.

These properties provide us with a powerful arsenal for simplifying expressions involving absolute values. By strategically applying these properties, we can break down complex expressions into simpler components and make the simplification process more manageable. In the context of our problem, the property |a - b| = |b - a| will be particularly useful in handling the terms |x - 7| and |7 - x|.

Step-by-Step Simplification of the Expression

Now that we have a solid understanding of absolute values and their properties, let's embark on the step-by-step simplification of the given expression:

∣x−7∣−∣7−x∣−∣2x−10∣∣5−x∣{ \frac{|x-7| - |7-x| - |2x-10|}{|5-x|} }

Step 1: Apply the Property |a - b| = |b - a|

We can rewrite the term |7 - x| as |x - 7| using the property |a - b| = |b - a|. This gives us:

∣x−7∣−∣x−7∣−∣2x−10∣∣5−x∣{ \frac{|x-7| - |x-7| - |2x-10|}{|5-x|} }

Step 2: Simplify the Numerator

Notice that we now have |x - 7| - |x - 7| in the numerator, which simplifies to 0. The expression becomes:

0−∣2x−10∣∣5−x∣{ \frac{0 - |2x-10|}{|5-x|} }

−∣2x−10∣∣5−x∣{ \frac{-|2x-10|}{|5-x|} }

Step 3: Factor out a Constant from the Absolute Value

We can factor out a 2 from the expression inside the absolute value in the numerator:

−∣2(x−5)∣∣5−x∣{ \frac{-|2(x-5)|}{|5-x|} }

Using the property |ab| = |a||b|, we get:

−2∣x−5∣∣5−x∣{ \frac{-2|x-5|}{|5-x|} }

Step 4: Apply the Property |a - b| = |b - a| Again

We can rewrite |5 - x| as |x - 5| in the denominator:

−2∣x−5∣∣x−5∣{ \frac{-2|x-5|}{|x-5|} }

Step 5: Consider the Cases Based on the Sign of x - 5

At this point, we need to consider two cases:

  • Case 1: x > 5

    If x > 5, then x - 5 > 0, so |x - 5| = x - 5. The expression becomes:

    −2(x−5)x−5{ \frac{-2(x-5)}{x-5} } = -2

  • Case 2: x < 5

    If x < 5, then x - 5 < 0, so |x - 5| = -(x - 5). The expression becomes:

    −2[−(x−5)]−(x−5){ \frac{-2[-(x-5)]}{-(x-5)} } = \frac{2(x-5)}{-(x-5)} = -2

Step 6: Exclude x = 5

It's crucial to note that the original expression is undefined when x = 5 because the denominator becomes zero. Therefore, our simplification is valid for all x ≠ 5.

Step 7: Conclusion

In both cases, the simplified expression is -2. Therefore, the correct option is:

C) -2

This step-by-step simplification demonstrates how we can effectively tackle expressions involving absolute values by applying the fundamental properties and considering different cases based on the sign of the expressions inside the absolute values. By carefully breaking down the problem into smaller, manageable steps, we arrived at the correct solution and gained a deeper understanding of the underlying concepts.

Choosing the Correct Option

Based on our step-by-step simplification, we have conclusively determined that the expression ∣x−7∣−∣7−x∣−∣2x−10∣∣5−x∣{ \frac{|x-7| - |7-x| - |2x-10|}{|5-x|} } simplifies to -2 for all x ≠ 5. Therefore, the correct option is:

C) -2

The other options are incorrect. Option A) 2 is the negation of the correct answer. Options B) 2x−14x−5{ \frac{2x-14}{x-5} } and D) 14−2xx−5{ \frac{14-2x}{x-5} } are more complex expressions that do not represent the simplified form of the given expression. Our systematic approach, utilizing the properties of absolute values and considering different cases, has led us to the accurate solution.

This exercise highlights the importance of careful and methodical simplification when dealing with absolute value expressions. By understanding the underlying principles and applying them strategically, we can navigate these problems with confidence and arrive at the correct answer.

Practical Applications of Absolute Value Simplification

While simplifying absolute value expressions may seem like a purely theoretical exercise, it has practical applications in various fields. Absolute values are used to represent magnitudes or distances, which are inherently non-negative quantities. This makes them indispensable in situations where we are concerned with the size of a quantity rather than its sign. Here are some examples of how absolute value simplification can be applied in real-world scenarios:

  1. Physics: In physics, absolute values are used to calculate the speed of an object, which is the magnitude of its velocity. For example, if an object has a velocity of -10 m/s (moving in the negative direction), its speed is |-10| = 10 m/s. Simplifying expressions involving absolute values can help physicists analyze motion and calculate other related quantities.

  2. Engineering: Engineers often deal with tolerances and deviations from ideal values. Absolute values are used to quantify these deviations, ensuring that designs meet specified requirements. For instance, if a component is designed to have a length of 10 cm with a tolerance of ±0.1 cm, the actual length must be within the range of 9.9 cm to 10.1 cm. This can be expressed using absolute values as |actual length - 10| ≤ 0.1. Simplifying expressions involving absolute values can help engineers analyze and control the variability in their designs.

  3. Computer Science: In computer science, absolute values are used in various algorithms and data structures. For example, in clustering algorithms, the distance between data points is often calculated using absolute values. Simplifying expressions involving absolute values can improve the efficiency of these algorithms.

  4. Economics: Economists use absolute values to measure the magnitude of changes in economic variables, such as inflation rates or GDP growth. For example, if the inflation rate changes from 2% to -1%, the magnitude of the change is |2 - (-1)| = 3 percentage points. Simplifying expressions involving absolute values can help economists analyze economic trends and make informed decisions.

  5. Everyday Life: Absolute values are also used in everyday situations. For example, when calculating the difference in temperature between two locations, we often use the absolute value to get a positive difference. If the temperature in one city is 25°C and in another city is 15°C, the temperature difference is |25 - 15| = 10°C. Simplifying expressions involving absolute values can help us make quick calculations and comparisons in our daily lives.

These examples illustrate that absolute value simplification is not just an abstract mathematical concept but a practical tool with wide-ranging applications. By mastering the techniques for simplifying absolute value expressions, we can enhance our problem-solving abilities and apply them to real-world scenarios.

Conclusion

In this comprehensive guide, we have explored the intricacies of simplifying the expression ∣x−7∣−∣7−x∣−∣2x−10∣∣5−x∣{ \frac{|x-7| - |7-x| - |2x-10|}{|5-x|} }. We began by understanding the fundamental concept of absolute values and their properties. We then systematically simplified the expression by applying these properties, considering different cases based on the sign of the expressions inside the absolute values, and arriving at the correct solution: -2.

We also discussed the practical applications of absolute value simplification in various fields, highlighting its relevance beyond the realm of pure mathematics. By mastering the techniques presented here, you are well-equipped to tackle similar challenges involving absolute values with confidence and apply your knowledge to real-world problems.

The journey through simplifying this expression has underscored the importance of a methodical approach, a solid understanding of fundamental concepts, and the ability to apply properties strategically. As you continue your mathematical explorations, remember that each problem is an opportunity to hone your skills and deepen your understanding. The principles we have discussed here will serve as a valuable foundation for tackling more complex mathematical challenges in the future.