Problem 38 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{15-|3-1|}{12-3 \cdot 2}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{13}{6}\).

Step 1: Evaluate the Absolute Value

First, calculate the expression inside the absolute value. The expression is \(|3-1|\). Perform the subtraction: \[3 - 1 = 2\]Then, find the absolute value, which is the non-negative difference:\[|2| = 2\]

Step 2: Simplify the Numerator

Replace the expression inside the numerator with the result from Step 1. So the numerator becomes:\[15 - 2\]Now, perform the subtraction:\[15 - 2 = 13\]

Step 3: Evaluate the Denominator

The denominator is \(12 - 3 \cdot 2\). According to the order of operations, perform the multiplication first:\[3 \cdot 2 = 6\] Then subtract the result from 12:\[12 - 6 = 6\]

Step 4: Simplify the Fraction

Now, simplify the fraction with the simplified numerator and denominator. The expression becomes:\[\frac{13}{6}\] Since 13 and 6 have no common factors other than 1, this fraction is in its simplest form.

Key Concepts

Understanding Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. Thus, the absolute value is always a non-negative number. In mathematical terms, the absolute value of a number \( x \) is denoted by \(|x|\). For example:

• The absolute value of 3, written as \(|3|\), is 3.
• The absolute value of -5, written as \(|-5|\), is 5.

This concept is crucial in simplifying expressions where the absolute value appears.
When faced with an absolute value in an expression, first solve any operations inside the absolute bars before applying the absolute value. For example, in the expression \(|3-1|\), first calculate the result of \(3 - 1\), which is 2.
Finally, the absolute value of 2 is simply 2.

The Importance of Order of Operations

Order of operations is a fundamental principle in math to ensure that expressions are consistently and correctly evaluated.
Known as PEMDAS/BODMAS, this order dictates:

• First, solve expressions inside Parentheses/Brackets.
• Next, evaluate Exponents or Orders.
• Then, perform Multiplication and Division, from left to right.
• Finally, perform Addition and Subtraction, from left to right.

This rule prevents confusion and mistakes in solving expressions. For example, in our problem, the denominator is \(12 - 3 \times 2\). According to the order of operations, you first perform the multiplication \(3 \times 2\) to get 6.
Then, subtract the result from 12 to arrive at 6.
Skipping or mixing up these steps can lead to incorrect results.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its smallest possible whole number numerator and denominator.
This process entails:

• Identifying any common factors between the numerator and denominator.
• Dividing both the numerator and denominator by their greatest common factor (GCF).

Simplified fractions are often easier to interpret and compare. In many cases, you'll find that fractions are already in simplest form, such as the fraction \(\frac{13}{6}\).
For \(\frac{13}{6}\), the numerator 13 and the denominator 6 share no common factors other than 1, so the fraction cannot be reduced further.
This shows that understanding when and why a fraction is in its simplest form is a key skill in algebra.