Problem 54 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y-4}-\frac{4}{4-y}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{y+4}{y-4} \)

Step 1: Identify the Denominators

The denominators are \(y - 4\) and \(4 - y\). These are opposite in sign, or additive inverses.

Step 2: Manipulate the Denominators to be Same

The subtraction sign before the second fraction can also be applied to the denominator, changing \(4-y\) to \(-(y-4)\). The expression becomes: \( \frac{y}{y-4} - \frac {4}{-(y-4)} \)

Step 3: Simplify the expression

Now the two denominators are the same, simplifying addition or subtraction of these fractions. After simplification, the expression becomes: \( \frac{y}{y-4} + \frac{4}{y-4} = \frac{y + 4}{y-4} \)

Key Concepts

Additive Inverses

Additive inverses are numbers or expressions that, when added together, result in zero. In the context of algebraic fractions, additive inverses help simplify expressions with opposite denominators. Consider two expressions, like \(a\) and \(-a\). Adding these gives:

• \(a + (-a) = 0\)

When dealing with fractions, recognizing additive inverses in denominators can transform complex operations like subtraction into simpler addition problems, as demonstrated in the example expression \(\frac{y}{y-4} - \frac{4}{4-y}\). Denominators \(y-4\) and \(4-y\) are inversely related. By applying the concept of additive inverses correctly, each can be adjusted to align for further simplification.

Fraction Simplification

Fraction simplification involves expressing a fraction in its simplest form by reducing the numerator and denominator to their smallest representations. This is achieved when they have no common factors other than one. Simplifying fractions can involve:

• Combining like terms
• Canceling identical factors across numerators and denominators

For instance, the fraction \(\frac{y}{y-4} + \frac{4}{y-4}\) can be combined by adding the numerators, leading to \(\frac{y+4}{y-4}\). This fraction is in its simplest form if the numerator cannot be factored further to cancel any terms with the denominator. Simplification makes calculations easier and results clearer.

Common Denominators

When adding or subtracting algebraic fractions, finding common denominators is key. A common denominator is a shared factor that allows integration of different fractions into a single entity. In an example like \(\frac{y}{y-4} - \frac{4}{4-y}\), each fraction initially has different, yet related, denominators. By identifying the denominators as additive inverses, they can be adjusted:

• The denominator \(4-y\) can be manipulated to \(-(y-4)\), matching the other fraction's denominator when negated.
• This results in \(\frac{y}{y-4} - \left(-\frac{4}{y-4}\right)\), transforming the operation into an addition.

Once common denominators are established, the fractions are combined more easily, leading to a straightforward solution. Identifying or creating common denominators streamlines the fraction addition and subtraction process significantly.