Problem 85 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ \frac{-2-5}{-7+(-7)} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{2} \).

Step 1: Simplify the Numerator

The given expression is \( \frac{-2-5}{-7+(-7)} \). First, simplify the numerator. The operation in the numerator \( -2 - 5 \) becomes \( -7 \). Thus, the simplified numerator is \( -7 \).

Step 2: Simplify the Denominator

Next, simplify the denominator. The operation in the denominator is \( -7 + (-7) \). This becomes \( -7 - 7 \), resulting in \( -14 \).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now, divide the simplified numerator by the simplified denominator. The expression becomes \( \frac{-7}{-14} \). Simplify \( \frac{-7}{-14} \) by dividing both the numerator and the denominator by \( -7 \), which results in \( \frac{-7 \div -7}{-14 \div -7} \). The result is \( \frac{1}{2} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra, allowing us to transform complex or lengthy expressions into more manageable forms. It's analogous to untangling a knot, where we work through each part step-by-step. In the context of algebraic fractions, this involves simplifying both the numerator and the denominator independently.

- Begin by addressing the operations in the numerator, then the denominator. This ensures clarity in each section before moving to the division step.- Always perform addition or subtraction operations within parentheses first. This keeps alignment with the order of operations (PEMDAS/BODMAS).- Simplifying expressions properly leads to clearer and more comprehensible results. This was evidenced in our given problem where simplifying \( -2 - 5 \) resulted in \( -7 \), streamlining the fraction's numerator.

Numerator and Denominator

In algebraic fractions, understanding the concepts of numerator and denominator is essential. The **numerator** is the top part of the fraction, while the **denominator** is the bottom part. Each part plays a significant role in the value of the fraction.

- The numerator tells us "how many" parts we have. In our problem, the numerator started as \( -2 - 5 \), reflecting a needed simplification to \( -7 \).- The denominator indicates "how many" of those parts make a whole. Initially, it was \(-7 + (-7)\), requiring simplification to \( -14 \).- Together, they represent a part-to-whole relationship in the fraction. By processing them separately, complex expressions like our example translate into simpler, actionable steps.

Division of Fractions

Performing division of fractions is central to solving algebraic expressions involving fractions. The division occurs once both the numerator and denominator are simplified to their basic forms.

- The fraction line itself implies division, essentially asking "how many groups of the denominator fit into the numerator."- In our example, dividing the simplified numerator \( -7 \) by the simplified denominator \( -14 \) involved adjusting the terms by their greatest common divisor.- Simplifying \( \frac{-7}{-14} \) by dividing both terms by \( -7 \) results in \( \frac{1}{2} \), a much more digestible value.- This underscores the importance of both simplification and division processes, making apparently complex expressions straightforward.