Problem 10 - Precalculus

Question

Evaluate each expression. $$|-3-4|-|-4|$$

Step-by-Step Solution

Answer

3

Step 1: Evaluate the first absolute value

The expression given is \(|-3-4|-|-4|\). Start by evaluating the first absolute value, which is \(|-3-4|\). This simplifies to \(|-7|\). The absolute value of -7 is 7.

Step 2: Evaluate the second absolute value

Next, evaluate the second absolute value in the expression, which is \(|-4|\). The absolute value of -4 is 4.

Step 3: Subtract the two evaluated absolute values

After calculating the absolute values, we have 7 and 4. The expression now is 7 - 4. Subtract 4 from 7 to get the final result, which is 3.

Key Concepts

Expression Evaluation

Expression evaluation is the process of simplifying mathematical expressions to find their value. In this exercise, we work with absolute values. The expression given is \(|-3 - 4| - |-4|\). Absolute value refers to the distance of a number from zero on a number line, regardless of the direction.

• To evaluate \(|-3 - 4|\), subtract within the absolute value first: \-3 - 4 = -7\.
• Then, find the absolute value of \-7\, which is \7\ because absolute values are always positive.
• For the second part, \|-4|\, its absolute value is \4\.

Now, subtract the two results: \7 - 4 = 3\. This step-by-step evaluation helps in breaking down expressions into simpler parts.

Precalculus

Precalculus serves as a bridge between algebra and calculus. It prepares students for the concepts they will encounter in calculus, such as limits and derivatives. Understanding absolute value is fundamental in precalculus because it helps with more complex functions and transformations.

• In precalculus, recognizing how absolute values can affect equations or inequalities is crucial.
• This concept is useful in solving problems involving distances or magnitudes, often used in real-world applications.
• Applying absolute value often implies two scenarios: one positive and one negative, and dealing with them requires logical structuring of expressions and equations.

Knowing how to evaluate absolute values in expressions lays groundwork for more advanced topics like piecewise functions and oscillating graphs.

Mathematics Education

Mathematics education aims to develop problem-solving and logical reasoning skills. Teaching students how to evaluate expressions like \(|-3 - 4| - |-4|\) is part of this educational journey.

• It highlights the importance of step-by-step logical thinking — breaking down each part of the expression into manageable pieces.
• Learning the concept of absolute value can also foster a deeper appreciation for mathematics as it applies to real-world situations, such as measuring distance.
• Providing students with both theoretical understanding and practical exercises enriches their learning experience.

Effective mathematics education ensures that learners are equipped to tackle complex problems and are prepared for more challenging mathematical concepts.