Problem 7 - Algebra for College Students

Question

Perform the indicated operations. Reduce answers to their lowest terms. See Example \(I\) $$ \frac{3 x}{2}+\frac{5 x}{2} $$

Step-by-Step Solution

Answer

The answer is \(4x\).

Step 1: Identify the Common Denominator

Notice that both fractions, \(\frac{3x}{2}\) and \(\frac{5x}{2}\), have the same denominator which is 2.

Step 2: Combine the Numerators

Since the denominators are the same, we add the numerators together: \(\frac{3x + 5x}{2}\).

Step 3: Simplify the Numerator

Combine the like terms in the numerator: \(3x + 5x = 8x\). This gives us the fraction: \(\frac{8x}{2}\).

Step 4: Reduce to the Lowest Terms

Simplify \(\frac{8x}{2}\) by dividing both the numerator and the denominator by 2: \(\frac{8x \div 2}{2 \div 2} = \frac{4x}{1} = 4x\).

Key Concepts

Common Denominator

When adding or subtracting fractions, the first step is to find a common denominator. A common denominator is a shared multiple of the denominators of two or more fractions. For example, in the problem \(\frac{3x}{2} + \frac{5x}{2}\), both fractions already have the common denominator of 2. This makes it easier to add them because we only need to focus on the numerators. If the denominators were different, we'd need to find the least common multiple (LCM) of the denominators to proceed.

Combine Like Terms

After finding a common denominator, the next step is to combine the numerators. In the exercise \(\frac{3x}{2} + \frac{5x}{2}\), the denominators are the same, so we add the numerators directly. This looks like: \(\frac{3x + 5x}{2}\). Combining like terms means adding or subtracting the coefficients of the same variable. Here, we add \(3x\) and \(5x\) resulting in \(8x\). Therefore, this step simplifies the fraction to \(\frac{8x}{2}\).

Simplify Fractions

Once the like terms are combined, the next step is to simplify the fraction if possible. Simplifying a fraction means making it as simple as possible. Take the new fraction from the problem: \(\frac{8x}{2}\). To simplify it, we see if there's a common factor in the numerator and the denominator. In this case, both 8 and 2 are divisible by 2. We perform the division: \(8x \div 2 = 4x\) and \(2 \div 2 = 1\). Thus, our simplified fraction becomes \(\frac{4x}{1}\).

Reduce to Lowest Terms

Finally, reducing a fraction to its lowest terms ensures the fraction is fully simplified. A fraction is in its lowest terms when no number other than 1 evenly divides both the numerator and the denominator. In the exercise, we simplified \(\frac{8x}{2}\) to \(\frac{4x}{1}\). \(\frac{4x}{1}\) is already in its lowest terms because the numerator cannot be divided by a number other than 1. Hence, the solution to \(\frac{3x}{2} + \frac{5x}{2}\) is \(4x\).