Problem 37 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(\frac{7}{18}\).

Step 1: Evaluate Absolute Value

First, evaluate the absolute value in the numerator. The expression inside the absolute value is \(6 - 2\). Calculate \(6 - 2 = 4\). Therefore, \(|6 - 2| = 4\).

Step 2: Simplify the Numerator

The simplified expression now is \(\frac{4 + 3}{8 + 2 \cdot 5}\). Add the numbers in the numerator: \(4 + 3 = 7\).

Step 3: Calculate the Sub-expression in the Denominator

Look at the denominator, and do the multiplication first according to the order of operations. Calculate \(2 \cdot 5 = 10\).

Step 4: Simplify the Denominator

Now, add \(8 + 10\) in the denominator: \(8 + 10 = 18\).

Step 5: Simplify the Fraction

The expression now is \(\frac{7}{18}\). Since 7 and 18 have no common factors other than 1, the fraction \(\frac{7}{18}\) is already in its simplest form.

Key Concepts

Absolute Value Simplification

When we talk about absolute value, we refer to how far a number is from zero on a number line, without considering direction. This means that both \( +5 \) and \( -5 \) have an absolute value of \( 5 \). As shown in the exercise, you start by finding the absolute value first because it's crucial for accurate calculations.

In the given problem, \( |6 - 2| \), you first subtract \( 2 \) from \( 6 \) which equals \( 4 \). Since \( 4 \) is already positive, its absolute value remains \( 4 \).

Remember:

• If the number inside the absolute value is positive, you simply get that number itself.
• If it's negative, you take away the negative sign to turn it positive.

By understanding this step thoroughly, you ensure that you are always working with correct values in any mathematical expression.

Order of Operations in Expressions

To solve expressions correctly, you must follow the order of operations, which is often remembered using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Following the order prevents calculation errors and ensures clear, accurate results. In the provided exercise:

• Numerator calculation was simplified by directly adding after handling the absolute value.
• Denominator operations started with multiplication, \( 2 \cdot 5 = 10 \), before addition, \( 8 + 10 = 18 \).

Adhering to this sequence is vital as it affects the outcome—any alteration can lead to incorrect answers. For example, failing to multiply before adding might result in wrong results. Make PEMDAS your guide to navigate through complex expressions with confidence!

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This implies that the numerator and denominator cannot be divided by any number other than \( 1 \). Simplified fractions are easier to interpret, compare, and use in further calculations.

After evaluating all parts of the expression, you end up with \( \frac{7}{18} \). Here's how this works:

• Check if both numbers share any common factors.
• If none exist besides \( 1 \) (as with \( 7 \) and \( 18 \)), the fraction is already in its simplest form.

Sometimes, simplifying fractions might involve dividing both top and bottom by the greatest common divisor (GCD). In this solution, since \( 7 \) is a prime number and doesn’t evenly divide \( 18 \), the fraction \( \frac{7}{18} \) remains unchanged. Understanding this process makes mathematical expressions precise and less cumbersome, forming a strong foundation for more advanced topics.