Problem 68 - Algebra for College Students

Question

Consider the addition problem \(\frac{8}{x-2}+\frac{5}{2-x}\). Note that the denominators are opposites of each other. If the property \(\frac{a}{-b}=-\frac{a}{b}\) is applied to the second fraction, we have \(\frac{5}{2-x}=-\frac{5}{x-2}\). Thus we proceed as follows: $$ \frac{8}{x-2}+\frac{5}{2-x}=\frac{8}{x-2}-\frac{5}{x-2}=\frac{8-5}{x-2}=\frac{3}{x-2} $$ Use this approach to do the following problems. (a) \(\frac{7}{x-1}+\frac{2}{1-x}\) (b) \(\frac{5}{2 x-1}+\frac{8}{1-2 x}\) (c) \(\frac{4}{a-3}-\frac{1}{3-a}\) (d) \(\frac{10}{a-9}-\frac{5}{9-a}\) (e) \(\frac{x^{2}}{x-1}-\frac{2 x-3}{1-x}\) (f) \(\frac{x^{2}}{x-4}-\frac{3 x-28}{4-x}\)

Step-by-Step Solution

Answer

(a) \( \frac{5}{x-1} \), (b) \( \frac{-3}{2x-1} \), (c) \( \frac{5}{a-3} \), (d) \( \frac{15}{a-9} \), (e) \( \frac{x^2 + 2x - 3}{x-1} \), (f) \( \frac{x^2 + 3x - 28}{x-4} \).

Step 1: Simplify Denominators (a)

Recognize that the denominators are opposites of each other. For \( \frac{2}{1-x} \), apply \( \frac{a}{-b}=-\frac{a}{b} \) to get \( -\frac{2}{x-1} \). Thus, \( \frac{7}{x-1} + \frac{2}{1-x} = \frac{7}{x-1} - \frac{2}{x-1} \).

Step 2: Combine Fractions (a)

Subtract the numerators over the common denominator: \( \frac{7-2}{x-1} = \frac{5}{x-1} \).

Step 1: Simplify Denominators (b)

For \( \frac{8}{1-2x} \), apply the property to get \( -\frac{8}{2x-1} \). Thus, \( \frac{5}{2x-1} + \frac{8}{1-2x} = \frac{5}{2x-1} - \frac{8}{2x-1} \).

Step 2: Combine Fractions (b)

Subtract the numerators over the common denominator: \( \frac{5-8}{2x-1} = \frac{-3}{2x-1} \).

Step 1: Simplify Denominators (c)

Apply the property to \( \frac{1}{3-a} \) to get \( -\frac{1}{a-3} \). Thus, \( \frac{4}{a-3} - \frac{1}{3-a} = \frac{4}{a-3} + \frac{1}{a-3} \).

Step 2: Combine Fractions (c)

Add the numerators over the common denominator: \( \frac{4+1}{a-3} = \frac{5}{a-3} \).

Step 1: Simplify Denominators (d)

For \( \frac{5}{9-a} \), apply the property to get \( -\frac{5}{a-9} \). Thus, \( \frac{10}{a-9} - \frac{5}{9-a} = \frac{10}{a-9} + \frac{5}{a-9} \).

Step 2: Combine Fractions (d)

Add the numerators over the common denominator: \( \frac{10+5}{a-9} = \frac{15}{a-9} \).

Step 1: Simplify Denominators (e)

Apply the property to \( \frac{2x-3}{1-x} \) to get \( -\frac{2x-3}{x-1} \). Thus, \( \frac{x^2}{x-1} - \frac{2x-3}{1-x} = \frac{x^2}{x-1} + \frac{2x-3}{x-1} \).

Step 2: Combine Fractions (e)

Add the numerators over the common denominator: \( \frac{x^2 + (2x-3)}{x-1} = \frac{x^2 + 2x - 3}{x-1} \).

Step 1: Simplify Denominators (f)

Recognize that \( \frac{3x-28}{4-x} \) can be rewritten using the property: \( -\frac{3x-28}{x-4} \). Hence, \( \frac{x^2}{x-4} - \frac{3x-28}{4-x} = \frac{x^2}{x-4} + \frac{3x-28}{x-4} \).

Step 2: Combine Fractions (f)

Add the numerators over the common denominator: \( \frac{x^2 + (3x-28)}{x-4} = \frac{x^2 + 3x - 28}{x-4} \).

Key Concepts

Opposite Denominators

When working with fractions, opposite denominators can sometimes appear, particularly in algebraic expressions. Opposite denominators look similar but have opposite signs. For example, consider the denominators in the fractions \( \frac{8}{x-2} \) and \( \frac{5}{2-x} \). Notice how we can rewrite \( \frac{5}{2-x} \) as \( -\frac{5}{x-2} \) by recognizing that \( 2-x \) is the opposite of \( x-2 \). This conversion uses the property \( \frac{a}{-b} = -\frac{a}{b} \), which allows us to standardize the denominator expressions. This crucial step enables further operations like addition or subtraction of fractions with ease and precision.
By converting to a common denominator, opposite denominator fractions are transformed into equivalent fractions with same denominators, thus simplifying the addition or subtraction process.

Fraction Addition

Adding fractions is one of the fundamental operations in dealing with rational expressions. When we add fractions, it's essential to have common denominators. This ensures that the fractions are aligned perfectly in terms of their parts. Once the denominators are the same, the numerators can be added directly.
Consider the expression \( \frac{8}{x-2} + \frac{5}{2-x} \). By utilizing opposite denominators strategy and rewriting \( \frac{5}{2-x} \) as \(-\frac{5}{x-2}\), both fractions now have a common denominator. This simplifies our work, and fraction addition becomes straightforward: \( \frac{8}{x-2} - \frac{5}{x-2} \). We simply add or subtract the numerators while keeping the denominator unchanged.

Simplifying Fractions

Once fractions are combined, it's important to simplify them if possible. Simplifying involves reducing the fraction to its simplest form. This means ensuring that the numerator and denominator have no common factors other than 1. Reducing fractions makes them easier to understand and work with in future calculations.
After performing addition or subtraction, such as in the case \( \frac{8-5}{x-2} \), we first perform the arithmetic operation on the numerators—getting \( 3 \) in this example. Once we have our new fraction \( \frac{3}{x-2} \), we check if it can be simplified further by factoring both numerator and denominator. If there are no further common factors, the fraction is already in its simplest form.

Numerators and Denominators

In fractions, numerators and denominators play distinct and vital roles. The numerator, the top number, represents the part of the whole that is being considered. The denominator, on the bottom, indicates the whole or the set that the numerator is a part of. Understanding their roles is crucial, particularly in algebraic fractions where expressions replace numbers.
When working with algebraic expressions, numerators and denominators can include variables and coefficients, such as in \( \frac{8}{x-2} \). Properly handling and combining these parts requires careful attention to their roles: numerators are added or subtracted, while denominators remain unchanged for the purpose of aligning the fractions.
The main goal is to ensure the integrity of the fractions during operations so that they accurately represent the intended values or expressions in an equation or algebraic fraction.