Problem 55 - Algebra and Trigonometry

Question

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 the Expression for (a)

We start with the expression for (a): \[| |-6|-|-4| |\]First, evaluate the absolute values inside the innermost brackets:\[|-6| = 6 \quad \text{and} \quad |-4| = 4\]Substitute these back into the expression:\[|6 - 4|\]Then, calculate the subtraction:\[|6 - 4| = |2|\]Finally, take the absolute value:\[|2| = 2\]So, the result of the expression for (a) is 2.

Step 2: Evaluate the Expression for (b)

Now consider the expression for (b): \[\frac{-1}{|-1|}\]First, calculate the absolute value in the denominator:\[|-1| = 1\]Substitute this value back into the expression:\[\frac{-1}{1} = -1\]Thus, the result of the expression for (b) is -1.

Key Concepts

Expression Evaluation

Expression evaluation involves assessing mathematical expressions by simplifying and applying known operations step-by-step. It is crucial to follow the correct order of operations to obtain the correct result. When evaluating expressions, especially those with absolute values, it's important to deal with the absolute value brackets first. Here’s a quick reminder of the basic rules:

• Start by simplifying the expression inside the absolute value brackets.
• Apply operations such as subtraction or division within the absolute value first.
• After operations within the absolute values are handled, simplify the expression further if needed.
• Finally, apply the absolute value, which always makes the result non-negative.

In the given exercise, for part (a), we began by solving the expression inside the absolute value bars before simplifying further. Simplify from the innermost part to the outer parts, respecting the absolute value rules throughout the process. By doing this systematically, we determined that the expression evaluates to 2.

Subtraction

Subtraction is one of the core arithmetic operations where you are essentially taking away or "subtracting" one number from another. When working with subtraction in an expression, it’s vital to pay attention to the numbers involved and handle any additional operations like absolute values correctly.
In our exercise, subtraction was used within absolute values. Initially, we simplified \( |6 - 4| \) by evaluating the difference between 6 and 4, resulting in 2. Here are some tips for handling subtraction effectively:

• Always subtract smaller numbers from larger numbers when in an absolute value to keep expressions non-negative.
• Remember that subtraction changes the direction on the number line; be careful with negative numbers.
• After performing subtraction, if the subtraction is within an absolute value, apply the absolute value to ensure the result is non-negative.

Understanding how to manage subtraction within absolute value context is key in various mathematical operations.

Division

Division is the process of splitting a number into equal parts. In an expression, division must be performed following any evaluation or simplification of absolute values. For normal division, remember that dividing by a positive number keeps the sign of the numerator, whereas dividing by a negative number changes the sign.
When dealing with an expression like the one in part (b) where you have \( \frac{-1}{|-1|} \):

• First, calculate the absolute value of the denominator.
• With \( |-1| = 1 \), substitute it back to get the expression: \( \frac{-1}{1} \).
• Now, perform the division: \( \frac{-1}{1} = -1 \).

It's crucial that you don’t overlook absolute values in the denominator, as they directly affect the final division outcome by turning the denominator into a positive value. This ensures accurate evaluation of the entire expression by maintaining the consistency of mathematical rules.