Problem 18 - Algebra 1

Question

Simplify the expression. $$\frac{9}{5 x}-\frac{2}{x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{9x-10}{5x^2}\).

Step 1: Identifying the denominators

First, identify that the denominators are \(5x\) and \(x^2\).

Step 2: Find the least common denominator

The least common denominator of \(5x\) and \(x^2\) is \(5x^2\).

Step 3: Rewrite fractions with common denominator

Next, rewrite the fractions to have the same common denominator. Multiply \(\frac{9}{5x}\) by \(\frac{x}{x}\) and \(\frac{2}{x^2}\) by \(\frac{5}{5}\) yielding \(\frac{9x}{5x^2}-\frac{10}{5x^2}\).

Step 4: Simplify the expression

Finally, simplify the expression by combining the fractions, which gives us \(\frac{9x-10}{5x^2}\).

Key Concepts

Least Common Denominator

When simplifying or combining fractions with different denominators, the first step is often finding the Least Common Denominator (LCD). Think of the LCD as the smallest number that both denominators can divide into without a remainder. In this way, you can "level the playing field," so to speak, allowing the fractions to be combined easily.
The denominators in the given exercise are \(5x\) and \(x^2\). To find the LCD, you need to consider the factors involved. The LCD must include each factor from each denominator, with the highest power of each appearing.Thus, for \(5x\) (which is \(5 \times x\)) and \(x^2\), the LCD is \(5x^2\).
This step involves a bit of practice, especially if the denominators look complicated. However, the key is to systematically evaluate the components (both numbers and variables) and combine them into a single, composite expression that accommodates both original denominators.

Combining Fractions

Once you've established a common denominator, the next step in working with fractions is to rewrite each fraction so that they share this denominator. This is crucial because you can only add or subtract fractions when their denominators match.
In our example, the fractions \(\frac{9}{5x}\) and \(\frac{2}{x^2}\) need to be rewritten with a common denominator of \(5x^2\). For \(\frac{9}{5x}\), multiply both the numerator and denominator by \(x\) to get \(\frac{9x}{5x^2}\). Similarly, for \(\frac{2}{x^2}\), multiply both by \(5\) to get \(\frac{10}{5x^2}\).
This transformation doesn't change the value of the fractions; it simply makes it possible to perform addition or subtraction. With denominators aligned, combining the numerators is straightforward, making it easy to add or subtract, as needed.

Rational Expressions

Rational expressions are similar to fractions but encompass polynomials in the numerator and the denominator. Simplifying them often requires the same skills as basic arithmetic fractions, but with an added emphasis on handling polynomials and variables.
When working with rational expressions like \(\frac{9x - 10}{5x^2}\), the goal is not just to combine terms but also to simplify further if possible by canceling out common factors or recognizing patterns like factoring or distributing.
This requires a keen eye on algebraic properties and often checking the solution for any restrictions or undefined points (like where the denominator could be zero). Such considerations ensure that not only is the expression simpler, but it is also accurate and applicable across all valid variable values.

• Identify potential factors in both the numerator and the denominator.
• Check for restrictions in the domain caused by potential zero-denominators.
• Always verify end results for completeness.

Understanding these elements helps in tackling more complex rational expressions with confidence and precision.