Problem 9 - Algebra for College Students

Question

Perform the indicated operations. Reduce answers to their lowest terms. See Example \(I\) $$ \frac{7 x}{2}-\frac{9 x}{2} $$

Step-by-Step Solution

Answer

-x

Step 1: Identify the common denominator

The fractions given are \(\frac{7x}{2}\) and \(\frac{9x}{2}\). Both fractions already have the same denominator, which is 2.

Step 2: Combine the numerators

Since the denominators are the same, we can combine the numerators directly. Subtract the numerators: \(\frac{7x - 9x}{2}\).

Step 3: Simplify the numerator

Simplify the expression in the numerator: \((7x - 9x) = -2x\). So, the fraction becomes \(\frac{-2x}{2}\).

Step 4: Reduce to lowest terms

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Thus, \(\frac{-2x}{2} = -x\).

Key Concepts

Common Denominator

When working with fractions, a common denominator is essential for performing addition or subtraction. In our exercise, we started with the fractions \(\frac{7x}{2}\) and \(\frac{9x}{2}\). Both fractions already have a common denominator of 2. This simplifies the process since we do not need to find a new common denominator. Having a common denominator allows us to focus on operations with the numerators directly.

Combine Numerators

Once you have a common denominator for the fractions you are working with, the next step is to combine the numerators. In our case, the fractions were \(\frac{7x}{2}\) and \(\frac{9x}{2}\). Since the denominators are the same, we can subtract the numerators directly: \(7x - 9x\). This results in \(\frac{7x - 9x}{2}\). By combining the numerators, we simplify the fractions, making further operations more manageable.

Simplify Fraction

After combining the numerators, we simplify the fraction as much as possible. From the previous step, we have \(\frac{7x - 9x}{2} = \frac{-2x}{2}\). Simplifying within the numerator \(7x - 9x = -2x\) gives us a simpler expression to work with. In our example, the fraction \(\frac{-2x}{2}\) was simplified by performing straightforward subtraction in the numerator.

Lowest Terms

Finally, we reduce the fraction to its lowest terms. This involves dividing both the numerator and the denominator by their greatest common divisor. For \(\frac{-2x}{2}\), both the numerator and the denominator have a greatest common divisor of 2. When we divide them by 2, we get \(\frac{-2x}{2} = -x\). Simplifying to the lowest terms ensures that our answer is as straightforward and easy to understand as possible.