Problem 42 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6 x+5}{x-2}+\frac{4 x}{2-x}$$

Step-by-Step Solution

Answer

The simplified result of adding these two fractions is \( \frac{10x+5}{x-2} \)

Step 1: Preparatory Step: Make Denominators the Same

Begin by making the denominators the same. Since the second fraction's denominator is negative, this equation could be rewritten as \( \frac{6x+5}{x-2} - \frac{-4x}{x-2} \). The negative sign before the second fraction is absorbed, which means the second fraction becomes positive.

Step 2: Add or Subtract the Fractions

Now that the fractions have the identical denominators, the numerators can be added or subtracted as directed. This will give: \( \frac{6x+5+4x}{x-2} = \frac{10x+5}{x-2} \)

Step 3: Simplify the Result

In the final step, the fraction will be simplified if possible. For this problem, it is impossible to simplify the result, as \(10x + 5\) and \(x - 2\) do not share a common divisor. Therefore, the simplified result of this fraction addition problem remains \( \frac{10x+5}{x-2} \).

Key Concepts

Additive Inverses

Understanding additive inverses is crucial when dealing with operations involving denominators. The additive inverse of a number is what you add to that number to get zero. For example, the additive inverse of 7 is -7, since 7 + (-7) = 0. This concept is helpful when working with fractions that have negative terms in either the numerators or denominators.

In the exercise given, we had to adjust the fractions so that their denominators appeared the same. The two original denominators were \( x-2 \) and \( 2-x \). You might notice that these two are negative versions of each other, or, practically speaking, additive inverses. Hence, by rewriting \( 2-x \) as \(-1 \cdot (x-2)\), we managed to carry that negative sign outside of the fraction. This adjustment helped us to proceed by adding or subtracting fractions effectively.

Simplifying Fractions

Simplifying fractions is an essential step in ensuring that your final answer is in its simplest form, making it easier to interpret.

• Simplification involves reducing a fraction to its lowest form, where the numerator and the denominator have no common divisors other than 1.
• To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

In our solution, after combining the fractions, we arrived at:\[ \frac{10x+5}{x-2} \]When analyzing \(10x+5\) and \(x-2\), it becomes clear that these two expressions do not share a common factor other than 1. Hence, further simplification isn't possible in this particular case. Always check if it’s possible to factor out any terms in both the numerator and the denominator, but remember that it’s not always feasible.

Common Denominators

When adding or subtracting fractions, having a common denominator is absolutely necessary. This means that the fractions should have the same bottom part, the denominator, which allows their numerators to be directly combined.Why is this important?

• A common denominator ensures that the pieces of the whole (fractions) we are dealing with are comparable.
• Without a common denominator, adding fractions by simply combining their numerators is not logically correct.

In the problem given, by identifying that \(x-2\) and \(2-x\) are additive inverses, we smartly transformed one fraction to have a matching denominator. This important move allowed us to seamlessly add the numerators and reach the final solution. Always remember to align your denominators before attempting any fraction operation!