Problem 44 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{9 x-1}{7 x-3}+\frac{6 x-2}{3-7 x}$$

Step-by-Step Solution

Answer

The simplified result of the exercise is \( \frac{3x + 1}{7x - 3} \)

Step 1: Recognise the Additive Inverses

Notice the denominators of both fractions are additive inverses of each other. \( 7x-3 \) and \( 3-7x \) are opposite in signs.

Step 2: Change the Signs

For the fraction with the negatively signed denominator, change the signs of both the numerator and the denominator. This results in : \( \frac{9x-1}{7x-3} - \frac{6x-2}{7x-3} \)

Step 3: Combine into One Fraction

Combine these two fractions since they now have a common denominator: \( \frac{9x - 6x - 1 + 2}{7x - 3} = \frac{3x + 1}{7x - 3} \)

Step 4: Simplify

The final simplified expression is \( \frac{3x + 1}{7x - 3} \)

Key Concepts

Additive Inverses

Understanding the concept of additive inverses is a fundamental skill in algebra, especially when dealing with algebraic fractions. The term 'additive inverse' refers to a number that, when added to a given number, results in zero. This is also often called the 'opposite' number. For example, the additive inverse of 7 is -7 because when they are added together (\(7 + (-7)\)), the sum is 0. In algebraic fractions, recognizing additive inverses can be useful for simplifying expressions.

Let's consider the original exercise: \[\frac{9 x-1}{7 x-3}+\frac{6 x-2}{3-7 x}\] Here, the denominators are additive inverses of each other (\(7x-3\text{ and }3-7x\text{ respectively}\)). To simplify, you first identify and then alter the signs to make them common denominators. This makes adding or subtracting fractions much easier, paving the way towards simplification.

Common Denominators

When adding or subtracting fractions, having common denominators is critical. A common denominator refers to a shared multiple of all the denominators involved and in most cases the least common denominator (LCD) is preferred. However, in the context of algebraic expressions where the denominators are polynomial expressions, common denominators are often created by manipulating signs or factoring.

In the exercise, the process involved changing the sign to transform the denominators into a common term. Once both fractions had the denominator \(7x-3\), they could be combined. This step is crucial as it paves the way for the addition or subtraction of the numerators, allowing us to move towards a single, simplified algebraic fraction.

Combining Like Terms

Combining like terms is a simplification process used to reduce and solve algebraic expressions. 'Like terms' refer to terms that have the same variable raised to the same power. Only the coefficients of like terms are added or subtracted while the variable part remains unchanged.

In our example, after establishing a common denominator, we combine the like terms in the numerators. The terms \(9x\) and \(6x\) are like terms because they both contain the variable \(x\) raised to the first power. So, we subtracted \(6x\) from \(9x\) to get \(3x\). Likewise, we combine the constant terms \(1\) and \(2\) by addition since they are also like terms, resulting in \(3\). The resulting simplified expression was \[\frac{3x+1}{7x-3}\].

Simplifying Expressions

Simplifying expressions involves reducing them to the simplest form while retaining their original value. The idea here is to combine like terms, factor where possible, and cancel terms to minimize the expression without altering its meaning.

In simplifying the given algebraic fractions, after obtaining common denominators and combining like terms, we were left with the expression \(\frac{3x + 1}{7x - 3}\). While this expression might not factor further or have terms that cancel out, it is in its simplest form. Simplifying expressions aids in understanding the underlying relationships in equations and makes further algebraic manipulations more manageable.