Problem 16 - Algebra 1

Question

Simplify the expression. $$\frac{9}{4 x}+\frac{7}{-5 x}$$

Step-by-Step Solution

Answer

-17/20x

Step 1: Find the Common Denominator

To add or subtract fractions, you must have the same denominator, also known as the common denominator. This can be found by multiplying the individual denominators together. Therefore, the common denominator for the two fractions \(\frac{9}{4x}\) and \(\frac{7}{-5x}\) is -20x.

Step 2: Rewrite the Fractions with the Common Denominator

Rewrite each fraction with the common denominator of -20x. Thus, \(\frac{9}{4x}\) becomes \(-\frac{9*5}{4x*-5}\) or, \(-\frac{45}{20x}\) and \(\frac{7}{-5x}\) becomes \(\frac{7*4}{-5x*4}\) or \(\frac{28}{20x}\).

Step 3: Perform the Addition

Now that the denominators are the same, add the numerators together to get \(-\frac{45 + 28}{20x}\) which simplifies to \(\frac{-17}{20x}\).

Key Concepts

Common Denominator

When simplifying or adding fractions, having a common denominator is crucial. The common denominator is essentially a shared base for all fractions involved in the expression, allowing you to perform additions or subtractions directly. In our example, we need to simplify \( \frac{9}{4x} + \frac{7}{-5x} \). The denominators here are \( 4x \) and \( -5x \). To find a common denominator, multiply these two denominators together to get \( -20x \).
This step is fundamental because it aligns the fractions in a comparable format, setting the stage for further operations. Remember, the multiplication of denominators means you find a common ground for both fractions without altering the underlying values.

Numerator Addition

Once fractions have a common denominator, you can proceed with adding their numerators. This step is straightforward but essential and requires careful attention to ensure accuracy. With our example fractions \( \frac{9}{4x} \) and \( \frac{7}{-5x} \) rewritten as \( -\frac{45}{20x} \) and \( \frac{28}{20x} \), thanks to our common denominator \( -20x \), we move to adding the numerators.

• Add the numerators: \( -45 + 28 \)
• Simplify the sum: \( -17 \)
• Keep the common denominator: \(-20x\)

Once summed, the result is \( \frac{-17}{20x} \). Notice that the operation only affects the numerators while the denominator remains consistent.

Algebraic Fractions

Algebraic fractions are fractions where the numerator or the denominator (or both) include algebraic expressions. Simplifying algebraic fractions involves similar techniques as numeric fractions, with additional care for algebraic terms that may include variables.
In our example of simplifying \(\frac{9}{4x} + \frac{7}{-5x}\), both denominators contain the variable \(x\). This adds a layer of complexity but follows the typical fraction rules. To simplify these correctly, one must not only find a common denominator but also accurately transform and simplify numerators which might include algebraic expressions. Algebraic simplification often requires factoring and recombining terms for streamlined expressions.
Mastering algebraic fractions can significantly boost your overall algebra skills, providing a foundation for tackling more advanced mathematical concepts. The key lies in keeping track of variables and ensuring every step aligns with both numerical and algebraic rules.