Problem 19 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

In Exercises 19-26, find the sum or difference. \(\frac{12}{5 x}-\frac{7}{6 x}\)

Step-by-Step Solution

Answer

\(\frac{1}{x}\)

Step 1: Identify the Common Denominator

The denominator for both fractions is \(x\). When the denominators are the same, fractions can be added or subtracted by adding or subtracting the numerators.

Step 2: Perform the subtraction

Subtract the numerator of the second fraction from the numerator of the first fraction while keeping the common denominator. This gives us: \(\frac{12}{5x} - \frac{7}{6x} = \frac{12}{5x} - \frac{7}{5x} = \frac{12-7}{5x} = \frac{5}{5x}\)

Step 3: Cancel out common terms

Reduce the fraction by cancelling out the 5 in the numerator and denominator. The result is \(\frac{1}{x}\)

Step 4: Conclusion

The sum of \(\frac{12}{5x} - \frac{7}{6x} = \frac{1}{x}\)

Key Concepts

Common Denominator

In the world of rational expressions, finding a common denominator is key to simplifying expressions, especially when performing addition or subtraction. A common denominator is a shared multiple of the denominators of two or more fractions. This shared denominator allows us to combine the fractions easily. Usually, when the fractions have differing denominators, we need to adjust them to have the same base for computation. However, in exercises like \[\frac{12}{5x} - \frac{7}{6x}\]it's important to recognize that though the coefficients of the denominators differ, the variable part, \(x\), is already common. This means there's no need to find another common base other than accommodating for those coefficients. Keeping the \(x\) consistent aids in performing smoother operations across these fractions.

Fraction Subtraction

Subtracting fractions might seem tricky initially, but it becomes easy with the right techniques. The principle is to align the fractions with a common denominator—which we've identified. Subtraction can then be approached in simple steps. Consider the fractions:\[\frac{12}{5x} - \frac{7}{6x}\]Ensure that the denominators match. Once you have a unified denominator, it's as simple as subtracting the numerators:

• Perform the subtraction: \(12 - 7\)
• Place the result over the common denominator: \(5x\)

This simplifies the expression to \(\frac{5}{5x}\). With this uniform base of \(5x\), we've successfully aligned the numerators for subtraction.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This step is crucial because it makes the expression easier to understand and work with. In our exercise, after performing subtraction we get \[\frac{5}{5x}\]To simplify, we need to identify any common factors in the numerator and the denominator. Here, both the numerator and the denominator share a "5." This means you can divide both by 5, which is known as "cancelling out common terms."

• Cancel the common factor: \(5\)
• The fraction then reduces to \(\frac{1}{x}\)

This process is fundamental in revealing the essence of the rational expression, which, in this case, is \(\frac{1}{x}\), thus completing the simplification.