Problem 39 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{|6-2|+3}{8+2 \cdot 5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{7}{18}\).

Step 1: Simplify the numerator

First, evaluate the absolute value in the numerator. The expression \(|6-2|\) evaluates to \(|4|\), which is equal to 4. Therefore, the numerator becomes \(4+3\).

Step 2: Finish simplifying the numerator

Add the numbers in the numerator: \(4+3=7\). So, the numerator is 7.

Step 3: Simplify the denominator

Next, evaluate the expression in the denominator. Start by calculating \(2 \cdot 5 = 10\).

Step 4: Finish simplifying the denominator

Add \(8 + 10\) to get 18. So the denominator is 18.

Step 5: Simplify the fraction

Combine the simplified numerator and denominator to get \(\frac{7}{18}\). Since 7 and 18 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics that students often encounter when dealing with algebraic expressions. The absolute value of a number is essentially the distance that number is away from zero on the number line. It is always a non-negative number.
For example, the absolute value of both -4 and 4 is 4 because they are both four units away from zero along the number line.
The notation for absolute value uses vertical bars: for example, |x| represents the absolute value of x.

• If x is a positive number or zero, |x| is just x. For instance, |5| = 5.
• If x is a negative number, |x| is the positive equivalent of that number. For example, |-5| = 5.

This essence of absolute value allows us to ignore any negative signs when simplifying expressions, focusing solely on the quantity itself without considering direction.

Numerator and Denominator Essentials

In any fraction, understanding the terms numerator and denominator is crucial. A fraction is composed of two parts: the numerator and the denominator. The numerator is the top part of the fraction, and the denominator is the bottom part.

• The numerator represents how many parts we have, or in other words, the number of parts taken.
• The denominator tells us the total number of equal parts the whole is divided into.

In our example, the expression starts as \( \frac{|6-2|+3}{8+2 \cdot 5}\). After simplifying, the numerator becomes 7 and the denominator becomes 18, resulting in the simplified fraction \( \frac{7}{18} \).
It’s important to recognize that despite mathematical operations performed on numerators and denominators, each has its own role in expressing the value of the fraction.

Simplifying Fractions

Simplifying fractions is a skill that makes complex expressions easier to handle and understand. The goal of simplifying fractions is to convert them into their simplest form, where the numerator and denominator have no common factors other than 1.
In our fraction \( \frac{7}{18} \), both the numerator (7) and the denominator (18) do not share any other common factors besides 1. This means \( \frac{7}{18} \) is already in its simplest form.

• To simplify a fraction, check for common factors in the numerator and denominator, and divide both by the greatest common factor.
• If no common factors exist other than 1, as in this case, then the fraction is already simplified.

Mastering the simplification of fractions can greatly ease calculations and provide a clearer understanding of the size or proportions the fraction represents.