Problem 100 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{|-3+6|+|-2+7|}{|-2 \cdot 2|}\)

Step-by-Step Solution

Answer

The evaluated expression is 2.

Step 1: Evaluate the Absolute Values Inside the Numerator

First, we need to evaluate each part inside the absolute values in the numerator.Calculate \(|-3+6| = |3| = 3\)and \(|-2+7| = |5| = 5\).

Step 2: Sum the Results in the Numerator

Now add the results from Step 1 to get the value of the numerator:\(3 + 5 = 8\).

Step 3: Evaluate the Absolute Value in the Denominator

Next, calculate the absolute value in the denominator:\(|-2 \cdot 2| = |-4| = 4\).

Step 4: Divide the Numerator by the Denominator

Finally, divide the result of the numerator by the denominator:\(\frac{8}{4} = 2\).

Key Concepts

Understanding Absolute Value

The concept of absolute value is fundamental when evaluating expressions. Absolute value refers to the magnitude of a number, ignoring its sign. It is always non-negative. Think of it as the distance from zero on a number line.
For any real number \( x \):

• If \( x \) is positive or zero, the absolute value is \( x \).
• If \( x \) is negative, the absolute value is \(-x \) to make it positive.

For example, in the expression \(|-3+6|\), calculate inside the absolute value first and find the absolute value of the result. The final value is \(|3| = 3\). This procedure applies to other components within absolute values like \(|-2+7|\), which converts to \(5\). Understanding this helps streamline calculations in expressions.

Role of the Numerator

In fractions, the numerator is the top part. It represents the number of parts we have out of a whole. When evaluating expressions, simplifying the numerator is crucial because it forms half of the expression.
In our example, we first simplified \(|-3+6|\) and \(|-2+7|\) to \(3\) and \(5\), respectively. Summing these gives us the numerator:\(3+5=8\).
This process ensures that you have the correct number of parts to be divided by the denominator, which is the next step. Always ensure the numerator is correctly calculated, as this impacts the final result of the division.

Importance of the Denominator

The denominator is the bottom part of a fraction. It indicates how many equal parts the whole is divided into. Correctly evaluating the denominator is crucial for correct division results.
In our exercise, the denominator \(|-2 \cdot 2|\) calculates first to \(-4\), and the absolute value changes it to \(4\). This process illustrates that the denominator needs to be a positive value to correctly express the number of parts in each whole unit you can extract from the numerator.
Errors in calculating the denominator can lead to incorrect results, so always pay close attention to calculations in this part of an expression.

Simplifying with Division

Division in mathematics involves distributing the numerator by the denominator. It is a critical part of solving and simplifying expressions.
In the given exercise, after obtaining \(8\) as the numerator and \(4\) as the denominator, the expression becomes \(\frac{8}{4}\). When you divide \(8\) by \(4\), the result is \(2\).
Understanding division means recognizing it as the process of determining how many times the denominator "fits into" the numerator. This process turns complex expressions into simplified results that are easier to interpret.