Problem 57 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{3 x-5}-\frac{5}{2 x+7} $$

Step-by-Step Solution

Answer

\(\frac{-x + 74}{(3x-5)(2x+7)}\)

Step 1: Find the Least Common Denominator (LCD)

To add or subtract rational expressions, we first need to find the least common denominator of the two fractions. Here, the denominators are \(3x - 5\) and \(2x + 7\). Since these are distinct, their least common denominator will be the product of the two, i.e., \((3x - 5)(2x + 7)\).

Step 2: Rewrite Each Fraction with the LCD

We rewrite each fraction so that both have the same denominator. Multiply the numerator and denominator of the first fraction, \(\frac{7}{3x-5}\), by \(2x+7\), and the numerator and denominator of the second fraction, \(\frac{5}{2x+7}\), by \(3x-5\). This gives us: \[\frac{7(2x+7)}{(3x-5)(2x+7)} - \frac{5(3x-5)}{(3x-5)(2x+7)}\]

Step 3: Expand the Numerators

Expand the numerators of both fractions from Step 2:For the first fraction:\(7(2x + 7) = 14x + 49\).For the second fraction:\(5(3x - 5) = 15x - 25\).The expression becomes:\[\frac{14x + 49}{(3x-5)(2x+7)} - \frac{15x - 25}{(3x-5)(2x+7)}\]

Step 4: Combine the Numerators

Now subtract the expanded numerators:\((14x + 49) - (15x - 25) = 14x + 49 - 15x + 25\)This simplifies to:\((14x - 15x) + (49 + 25) = -x + 74\).The expression is now:\[\frac{-x + 74}{(3x-5)(2x+7)}\]

Step 5: Check for Further Simplification

Verify if the simplified numerator \(-x + 74\) shares any common factors with the denominator \((3x-5)(2x+7)\). Since there are no common factors, the expression \(\frac{-x + 74}{(3x-5)(2x+7)}\) is in its simplest form.

Key Concepts

Least Common Denominator

When working with rational expressions, such as fractions with variables, finding the Least Common Denominator (LCD) is crucial. This step depends on the denominators of the fractions you're working with. The least common denominator is the smallest expression that both denominators can divide into without a remainder. To find the LCD when the denominators are completely different, like in this exercise with denominators \(3x - 5\) and \(2x + 7\), multiply them together: \((3x - 5)(2x + 7)\). This product becomes the new denominator for both rational expressions, allowing you to add or subtract them with ease.

• Multiply the distinct denominators to find the LCD.
• Use the LCD to rewrite each fraction.

Remembering these steps helps in smoothly handling rational expressions in algebra.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra, especially when dealing with rational expressions. In our exercise, once the fractions have the same denominator, our job is to simplify the numerators before subtracting them. We do this by expanding the products:

• For the first fraction, multiplying gives \(7(2x + 7) = 14x + 49\).
• For the second fraction, calculate \(5(3x - 5) = 15x - 25\).

Looking at the numerators after expansion allows us to combine them, subtracting one from the other, while aligning the fractions under their common denominator. The expression becomes simpler, as we manage the terms separately before combining them at the end. After combining, further check if the resulting numerators and denominators can be simplified by finding common factors. If none exist, as in our example \(-x + 74\) does not share factors with \((3x-5)(2x+7)\), it stands fully simplified.

Algebraic Fractions

Algebraic fractions, also known as rational expressions, resemble regular fractions but contain polynomials in their numerator and denominator rather than just numbers. Understanding their behavior, especially how to manage operations involving them, is a fundamental skill in algebra. These fractions require careful attention to the variables and signs involved.

• They expand our ability to describe relationships and operations in a more general form.
• They require techniques such as factoring, finding common denominators, and checking simplifications.

In our particular exercise, algebraic fractions teach us to treat variable terms with the same respect as numerical terms, paying close attention to their manipulation in both the numerator and denominator. This careful handling leads to better accuracy in deriving final expressions, refining your algebraic problem-solving proficiency.