Problem 39 - A Graphical Approach to College Algebra

Question

Which of these rational expressions equal \(-1 ?\) (Assume that all denominators are nonzero. A. \(\frac{x-4}{x+4} \quad\) B. \(\frac{-x-4}{x+4}\) C. \(\frac{x-4}{4-x} \quad\) D. \(\frac{x-4}{-x-4}\)

Step-by-Step Solution

Answer

Expressions B and C equal \(-1\).

Step 1: Analyze Expression A

Let's simplify the expression \( \frac{x-4}{x+4} \). There is no manipulation that will change this to \(-1\), since neither numerator nor denominator is a negative counterpart of each other: \(x-4 eq -(x+4)\). So, this expression does not equal \(-1\).

Step 2: Analyze Expression B

Consider the expression \( \frac{-x-4}{x+4}\). Simplify the numerator: \(-x-4\) which can be rewritten as \(-(x+4)\). This shows that \(\frac{-x-4}{x+4} = \frac{-(x+4)}{x+4} = -1\). Therefore, expression B equals \(-1\).

Step 3: Analyze Expression C

For the expression \( \frac{x-4}{4-x} \), notice that \(4-x\) is the same as \(-(x-4)\). This means \(\frac{x-4}{4-x} = \frac{x-4}{-(x-4)} = -1\). Hence, expression C is equal to \(-1\).

Step 4: Analyze Expression D

The expression \( \frac{x-4}{-x-4}\) can be rewritten by manipulating the denominator as \(-x-4 = -(x+4)\). So, we have \(\frac{x-4}{-(x+4)}\), which is not \(-1\) because the numerator \(x-4 eq -(x+4)\). Thus, expression D does not equal \(-1\).

Step 5: Final Step: Conclusion

From the analysis, expressions B and C equal \(-1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can combine numbers, variables, and operators (like plus, minus, multiplication, and division). They can be as simple as a single number or more complex with multiple variables and operations. For example, in the context of our exercise, the rational expressions such as \( \frac{x-4}{x+4} \) or \( \frac{-x-4}{x+4} \) are forms of algebraic expressions because they contain variables “x” and arithmetic operations like subtraction and division.
Understanding algebraic expressions is crucial because they are fundamental in many areas of mathematics and help us to model real-world situations mathematically. When working with these expressions, we often apply rules to simplify or manipulate them in order to solve equations, which is what the original exercise required.
It’s important to note, while dealing with rational expressions, that the denominator should not be zero because division by zero is undefined. Therefore, we assume all denominators in our exercise are nonzero to keep the expressions valid.

Simplification

Simplification is the process of making an algebraic expression easier to work with by reducing it to its simplest form. In rational expressions, this often involves factoring out common parts or using algebraic identities to express them more clearly.
In our step-by-step solution, we simplified the numerators and denominators to determine if they equaled \(-1\). For example, in Expression B, the numerator \(-x-4\) simplifies to \(-(x+4)\), allowing us to easily see how it cancels out the same components in the denominator, resulting in the value \(-1\).
The ability to simplify rational expressions plays a key role in solving equations because it allows us to see relationships between terms more clearly. It also aids in comparing rational expressions like we did in the exercise, making it easier to identify equivalent expressions.

Negative Numbers

Negative numbers are an essential part of algebra and mathematics as a whole. They are numbers less than zero, represented with a minus sign. In rational expressions, a negative sign can often be factored out, as seen in our exercise.
For instance, in Expression C, we observed that \(4-x\) is equivalent to \(-(x-4)\). This highlights one of the fundamental rules for negative numbers: changing the order of subtraction results in a negative sign, i.e., \(a-b = -(b-a)\). Understanding this property is critical when simplifying and comparing such terms.
When working with negative numbers, always pay attention to how they affect the expressions, especially during operations like simplification or factorization. This awareness can clarify whether an expression results in a negative or positive value, which is essential for correctly solving problems or comparing rational expressions as we did to find which ones equaled \(-1\).