Problem 50 - Elementary Algebra

Question

Determine each of the values, \(|-(46-|-24|)|\)

Step-by-Step Solution

Answer

Answer: 22

Step 1: Evaluate the inner absolute value

To evaluate \(|-24|\), Recall that the absolute value of a number is its distance from 0 on the number line, regardless of direction. So, \(|-24| = 24\)

Step 2: Substitute the result back into the expression

Now let's replace \(|-24|\) with 24 in the main expression: \(|-(46-|-24|)| = |-(46-24)|\)

Step 3: Evaluate the subtraction inside the absolute value

Perform the subtraction inside the absolute value: \(|-(46-24)| = |-22|\)

Step 4: Evaluate the final absolute value

Evaluate the absolute value of -22: \(|-22| = 22\) The value of \(|-(46-|-24|)|\) is 22.

Key Concepts

Elementary Algebra

At its core, elementary algebra involves the study of mathematical symbols and the rules for manipulating these symbols in formulas. It's a unifying thread across almost all of mathematics and provides a foundation for more advanced topics in the subject. In our example, algebraic expressions and operations come into play. When we evaluate the expression \(|-(46-|-24|)|\), we are dealing with variables and constants, operations such as subtraction, and the concept of absolute value.

Understanding how to work with absolute values, which measure the distance of a number from zero, is a cornerstone in algebra. The 'distance' concept means that no matter if the value is negative or positive, its absolute value will always be a positive number or zero. Applying this concept allowed us to transform \(|-24|\) into 24, simplifying the expression step by step.

Number Line

The number line is a visual representation of numbers laid out in a straight line where each point corresponds to a number. This tool is incredibly helpful in understanding the concept of absolute value. On a number line, the distance of a number from zero is represented by its absolute value. For instance, both -24 and +24 are 24 units away from zero. So, their absolute value is 24, regardless of the direction.

When we look at \(|-24|\), it doesn't matter that -24 is on the left side of zero on the number line; its absolute value is still 24 units. Remember that on a number line, left indicates negative, right indicates positive, and the distance to zero is always a non-negative number. Visualizing this can help students grasp why the absolute value of a negative number is positive.

Order of Operations

Also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), the order of operations is a set of rules that dictate the sequence in which parts of a mathematical expression should be solved to ensure consistency in the final results. For our example, the innermost absolute value is addressed first, followed by subtraction within the parentheses.

Let's revisit our steps:

• First, evaluate the inner absolute value \(|-24|\).
• Then, perform the subtraction in the parentheses \((46-24)\).
• Finally, address the outer absolute value the last \(|-22|\).

By following this order of operations, we can systematically work through the problem, and ensure the correctness of our solution, arriving confidently at 22.