Problem 63 - Algebra and Trigonometry

Question

\(61-66=\) Evaluate each expression. $$ \begin{array}{ll}{\text { (a) }| |-6|-|-4| |} & {\text { (b) } \frac{-1}{|-1|}}\end{array} $$

Step-by-Step Solution

Answer

(a) 2, (b) -1.

Step 1: Evaluate Inner Absolute Values in (a)

For the expression \(| |-6|-|-4| |\), begin by evaluating the absolute values inside first. The absolute value of \(-6\) is \(|-6| = 6\) and the absolute value of \(-4\) is \(|-4| = 4\).

Step 2: Evaluate Outer Absolute Values in (a)

Replace the inner absolute values with their calculated results: \(| 6 - 4 |\). Next, perform the subtraction: \(6 - 4 = 2\). Then take the absolute value of the result, \(| 2 | = 2\).

Step 3: Evaluate Absolute Value in (b)

For the expression \(\frac{-1}{|-1|}\), evaluate the absolute value in the denominator first. The absolute value of \(-1\) is \(|-1| = 1\).

Step 4: Simplify (b) with Calculated Absolute Value

Now substitute the calculated absolute value into the expression: \(rac{-1}{1}\). Perform the division to get \(-1\).

Key Concepts

Operations with Absolute Values

When dealing with absolute values, we are often tasked with evaluating expressions using various operations. The absolute value of a number is simply its distance from zero on the number line and is always non-negative. While performing operations like addition, subtraction, multiplication, or division with absolute values, it's important to handle the absolute value function properly.

• Start by evaluating any absolute value expressions by removing the negative sign if it exists. For example, \(|-3| = 3\).
• Complete any operations inside the absolute value symbols first, before proceeding to the outer operations.
• Manage the order of operations by solving expressions within parentheses first, followed by absolute values, and then any other operations.

Understanding these steps ensures that each operation is performed accurately, maintaining the integrity of the mathematical expression.

Absolute Value Evaluation

Evaluating absolute values requires understanding that you're determining the non-negative value of any given number. This is indicated by two vertical bars, such as \(| x |\). Let's consider the evaluation process used in the exercise:

• For the number \(-6\), the absolute value is \(|-6| = 6\); for \(-4\), it becomes \(|-4| = 4\).
• The expression \(| 6 - 4 |\) emphasizes resolving what's inside first before applying the absolute value. You evaluate the expression normally by subtracting \(6 - 4 = 2\), and then calculate the absolute value of 2, which remains \(2\).
• This process is illustrated again in part (b) of the problem, where the absolute value of \(-1\) is \(|-1| = 1\), simplifying the expression down to a standard operation.

Taking the absolute value is about distancing the number from zero, so keep it positive, unless the context of the problem changes. It's one of the simplest yet crucial parts of correctly interpreting and evaluating expressions.

Subtraction with Absolute Values

Subtracting with absolute values can slightly differ from regular subtraction because the involvement of absolute values might change the order in how operations are performed. To elucidate this concept:

• Begin by evaluating any expressions within the absolute value symbols. This was demonstrated with \(| |-6|-|-4| |\) in the exercise, where initial subtraction \(6 - 4\) happens after evaluating absolute values of \(-6\) and \(-4\).
• Once on the outside of absolute values, perform the subtraction. This interior calculation \(6 - 4\) resolves to \(2\), where subsequently the absolute value makes sure the result does not change signs.
• The subtraction across the absolute value doesn't influence the calculation, as any result is taken as non-negative, ensuring no negative results impact the overall operation.

By understanding subtraction implications within absolute values, it allows for more accuracy in calculating expressions and avoiding potential sign errors.