Problem 40 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{4 x}{5 x-2}-\frac{2 x}{5 x+1} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2(x+0.8)}{(x-0.4)(x+0.2)}\).

Step 1: Identify the common denominator

The common denominator of the fractions \(\frac{4x}{5x-2}\) and \(\frac{2x}{5x+1}\) is the product of their denominators, which is \((5x-2)(5x+1)\).

Step 2: Rewrite the fractions with the common denominator

Rewrite the fractions as \(\frac{4x(5x+1)}{(5x-2)(5x+1)}-\frac{2x(5x-2)}{(5x-2)(5x+1)}\) by multiplying the numerator and the denominator of the first fraction by \(5x+1\) and of the second fraction by \(5x-2\).

Step 3: Subtract the fractions

Subtract the fractions to get \(\frac{4x(5x+1)-2x(5x-2)}{(5x-2)(5x+1)}\).

Step 4: Simplify the numerator

Simplify the numerator to get \(\frac{20x^2+4x-10x^2+4x}{(5x-2)(5x+1)}\).

Step 5: Simplify the final result

Simplify to get \(\frac{10x^2+8x}{(5x-2)(5x+1)}\). Simplifying further, we get \(\frac{10x(x+0.8)}{(5x-2)(5x+1)}\). Dividing both numerator and denominator by 5x, we get \(\frac{2(x+0.8)}{(x-0.4)(x+0.2)}\).

Key Concepts

Common Denominator

One of the key steps in subtracting fractions is to determine a common denominator. This is essential because fractions can only be combined (added or subtracted) when they have the same denominator. Finding a common denominator involves identifying a number or an expression that both denominators can divide into evenly. In the case of algebraic expressions, this usually means multiplying the unique factors of each denominator together.

For example, when working with the expression \(\frac{4x}{5x-2} - \frac{2x}{5x+1}\), we recognize that the denominators are \(5x-2\) and \(5x+1\). These two expressions do not share any common factors, so the common denominator will be their product: \( (5x-2)(5x+1) \). By establishing this common denominator, it creates a common ground that allows for the direct subtraction of the numerators.

Subtracting Fractions in Algebra

Subtracting fractions within algebraic contexts follows the same principles as subtracting numerical fractions, but with the inclusion of variables. Once a common denominator has been established, the next step is to express each fraction with this common denominator. We do this by multiplying the numerator and denominator of each fraction by whatever is needed to make the denominators match.

In our example with \(\frac{4x}{5x-2}\) and \(\frac{2x}{5x+1}\), we adjust the fractions as such: \(\frac{4x(5x+1)}{(5x-2)(5x+1)} - \frac{2x(5x-2)}{(5x-2)(5x+1)}\). Notice how each fraction is transformed to have the common denominator without changing the value of the fraction itself. Once the fractions have the same denominator, we can then proceed to subtract the numerators, combining like terms where possible, to achieve a single simplified expression.

Simplify Numerator

After aligning the denominators and subtracting the algebraic fractions, we must simplify the numerator to reach the most reduced form of the expression. Simplifying the numerator is done by expanding any multiplied factors, combining like terms, and reducing expressions where possible.

Following the previous steps, we arrived at \(\frac{4x(5x+1) - 2x(5x-2)}{(5x-2)(5x+1)}\). When we simplify the numerator, we expand the products to get \(\frac{20x^2 + 4x - 10x^2 + 4x}{(5x-2)(5x+1)}\). Combining like terms gives us \(\frac{10x^2 + 8x}{(5x-2)(5x+1)}\), which is simpler but may not be fully reduced. If possible, factor out common factors in the numerator and denominator to further simplify. In this case, we can factor out a \(5x\) from both, yielding \(\frac{2(x+0.8)}{(x-0.4)(x+0.2)}\). The expression is now in its simplest form, allowing for easier interpretation and further manipulation if required.