Problem 67 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Simplify the Numerator

We start by simplifying the numerator of the expression \(-2 - 5\).\(-2 - 5 = -7\). Thus, the numerator simplifies to \(-7\).

Step 2: Simplify the Denominator

Next, we simplify the denominator of the expression \(-7 + (-7)\). This is the same as \(-7 - 7 = -14\). Thus, the denominator simplifies to \(-14\).

Step 3: Divide Numerator by Denominator

Now, we divide the simplified numerator by the simplified denominator to evaluate the expression:\(\frac{-7}{-14}\). Dividing \(-7\) by \(-14\) gives us \(\frac{1}{2}\), because a negative divided by a negative results in a positive value.

Key Concepts

Numerator Simplification

Simplifying the numerator is the first step to making any fraction simpler. In the context of the given exercise, the numerator begins with an expression \(-2 - 5\). When handling negative numbers, remember that subtracting makes the values more negative. Thus, \(-2\) minus \(5\) becomes \(-7\). To simplify the numerator:

• Identify all terms present in the numerator.
• Apply basic arithmetic operations, taking note of positive and negative signs.

In our example, we only needed to consider two numbers, which made simplification straightforward. Once complete, your numerator will be ready for the next step in fraction simplification.

Denominator Simplification

Next, you must simplify the denominator. In this exercise, the denominator consists of the expression \(-7 + (-7)\). Adding two negative numbers is like subtracting their absolute values from zero. Since both numbers are \(-7\), combining them gives \(-14\).Steps to simplify the denominator:

• Recognize all components in the denominator.
• Combine similar terms while respecting their signs (positive or negative).

In practice, attend to the signs carefully. They determine the arithmetic action you'll take: addition or subtraction. Simplifying the denominator ensures the fraction is set for final computation.

Division of Integers

The final crucial step is dividing the simplified numerator by the simplified denominator. In your fraction, both the numerator and the denominator are negative: \(\frac{-7}{-14}\). Dividing two negative numbers entails that the resulting quotient becomes positive. Hence, \(-7\) divided by \(-14\) results in \(\frac{1}{2}\).For dividing integers:

• Identify the signs. If both integers share the same sign, the result is positive.
• If the signs differ, the result is negative.

Always simplify the resulting fraction if possible, though in this particular problem, \(\frac{1}{2}\) is already in its simplest form. This process wraps up the simplification of a fraction involving division of integers.