Problem 86 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ \frac{-3-(-1)}{-2+(-2)} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Numerator

The numerator of the expression is \(-3 - (-1)\). We simplify this by recognizing that subtracting a negative is the same as adding the positive. So we rewrite the expression as \(-3 + 1\). Calculate this to get \(-2\).

Step 2: Simplify the Denominator

The denominator of the expression is \(-2 + (-2)\). This can be simplified by recognizing that adding a negative is equivalent to subtraction. Therefore, we rewrite it as \(-2 - 2\). Calculate this to get \(-4\).

Step 3: Simplify the Fraction

Now that the numerator is \(-2\) and the denominator is \(-4\), the expression becomes \(\frac{-2}{-4}\). We know that a negative divided by a negative yields a positive result. Simplify \(\frac{-2}{-4}\) to \(\frac{1}{2}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

Key Concepts

Numerator and Denominator

In any rational expression or fraction, the individual constituents, namely the numerator and the denominator, play pivotal roles. The numerator is the number or expression above the division line, while the denominator resides below it. Together, these two form a fraction: \( \frac{\text{numerator}}{\text{denominator}} \). Understanding how they interact is crucial in mathematical problem-solving.

• **Numerator:** In our example, this is \(-3 - (-1)\). It's what you want to divide.
• **Denominator:** Here, this is \(-2 + (-2)\). It's the number you divide by.

These elements allow us to understand what is being divided and how it affects the overall value of the fraction. In our exercise, simplifying both parts separately helps break down the problem into smaller, more manageable steps.

Simplification

Simplification is the process of reducing an expression to its simplest form. It's an integral part of solving expressions involving fractions. For the rational expression \[\frac{-3-(-1)}{-2+(-2)}\], simplification involves working through both the numerator and the denominator independently, and then simplifying any remaining fraction.

• **Numerator Simplification:** \(-3 - (-1)\) transforms into \(-3 + 1\), simplifying further to \(-2\).
• **Denominator Simplification:** \(-2 + (-2)\) becomes \(-2 - 2\), resulting in \(-4\).
• **Fraction Simplification:** When both parts have been individually simplified as \(-2\) over \(-4\), you handle the fraction by dividing each term by \(2\), reducing the fraction to \(rac{1}{2}\).

Simplification often makes calculations easier and helps in revealing the true value of the expression being evaluated.

Fractions

A fraction signifies a part of a whole and is one of the fundamental concepts in mathematics. They are represented as \(\frac{a}{b}\) where \(a\) is the numerator and \(b\) is the denominator.
Some key points about fractions:

• **Proper Fractions:** The numerator is less than the denominator, such as \(rac{1}{2}\).
• **Improper Fractions:** The numerator is greater than or equal to the denominator, like \(rac{4}{3}\).
• **Mixed Numbers:** These are combinations of an integer and a proper fraction.
• **Equivalent Fractions:** No matter the form, equivalent fractions have the same value. This is why simplifying \(rac{-2}{-4}\) to \(rac{1}{2}\) is valid.

Fractions require understanding for basic operations like addition, subtraction, multiplication, and division to be performed accurately, incorporating broader mathematical principles.