Problem 56 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \begin{array}{ll}{\text { (a) }|2-|-12| |} & {\text { (b) }-1-|1-|-1| |}\end{array} $$

Step-by-Step Solution

Answer

(a) 10, (b) -1.

Step 1: Identify the Absolute Value

In part (a), first evaluate the innermost absolute value: \(|-12|\). The absolute value of \(-12\) is \(12\), because absolute value converts negative numbers to their positive counterparts.

Step 2: Substitute the Evaluated Absolute Value

Substitute \(|-12|\) with \(12\) in the expression \(|2 - |-12|\). This changes the expression to \(|2 - 12|\).

Step 3: Evaluate the New Expression

Calculate \(2 - 12\), which equals \(-10\).

Step 4: Apply Absolute Value Again

Apply absolute value to \(-10\): \(|-10| = 10\). Thus, the value of part (a) is \(10\).

Step 5: Evaluate Innermost Absolute Value in Part (b)

Start with the innermost absolute value in part (b): \(|-1|\), which is \(1\).

Step 6: Substitute the Evaluated Absolute Value in Part (b)

Replace \(|-1|\) with \(1\) in the expression \(-1 - |1 - |-1|\), leading to \(-1 - |1 - 1|\).

Step 7: Calculate Inside the Absolute Value

Simplify inside the last absolute value by calculating \(1 - 1\), resulting in \(0\).

Step 8: Apply Absolute Value to Zero

The absolute value of zero is \(0\); therefore, replace \(|1 - 1|\) with \(0\), giving the expression \(-1 - 0\).

Step 9: Final Evaluation

Compute the expression \(-1 - 0\) which results in \(-1\). Thus, the value of part (b) is \(-1\).

Key Concepts

Integer Operations

Integer operations are basic arithmetic actions involving integers, which are whole numbers that can be positive, negative, or zero. When working with integer operations, keeping track of positive and negative signs is crucial.

• Addition and Subtraction: When adding or subtracting integers, consider the signs. For example, adding two positive integers yields a positive result (e.g., \(3 + 5 = 8\)), while adding two negative integers results in a larger negative value (e.g., \(-3 + -5 = -8\)). Subtraction can be seen as adding a negative (e.g., \(5 - 3 = 5 + (-3)\)).
• Multiplication and Division: When multiplying or dividing two integers, if both integers have the same sign, the result is positive. Conversely, if the signs differ, the result is negative (e.g., \((-4) \times 2 = -8\)). Remember, any number multiplied or divided by zero remains zero.

Mastering integer operations is fundamental as they form the building blocks for more complex expressions and equations.

Number Line

A number line is a visual representation of numbers on a straight line, where each point on the line corresponds to a number. It extends infinitely in both directions—a useful tool for understanding integers and operations like absolute value and subtraction.

• Understanding Position: On a number line, numbers increase to the right and decrease to the left. Zero is the central point separating positive numbers from negatives.
• Using Number Lines for Integer Operations: For addition, move to the right if adding a positive number and to the left for a negative number. For subtraction, do the reverse: move left for positive and right for negative.
• Visualizing Absolute Value: The absolute value of a number can be seen as its distance from zero on the number line. It's always non-negative because distance cannot be negative.

Using the number line effectively helps in visualizing and solving mathematical problems involving integers and absolute values.

Calculating Expressions

Calculating expressions involves performing mathematical operations to find a numerical value. It requires understanding the rules of operations and the order in which they should be performed.

• Order of Operations: Follow the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication/Division, Addition/Subtraction). This rule dictates the sequence in which calculations should be done.
• Working with Absolute Values: Begin by solving expressions inside the absolute value sign. Compute any operations within it using standard order, then apply the absolute value operation by converting negative results to positive.
• Tracking Your Steps: Break complex expressions into smaller parts. Tackle one item at a time, solving each before progressing. This method reduces errors and keeps calculations organized.

By mastering the process of calculating expressions with integer operations and absolute values, solving even complicated expressions becomes much easier and manageable.