Problem 63 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}-2}{x^{2}+6 x-7}+\frac{19-4 x}{7-6 x-x^{2}}$$

Step-by-Step Solution

Answer

The simplified result of the given expression is \( \frac{x^{2}-4x-21}{(x-1)(x+7)} \)

Step 1: Simplify the denominators and numerators

Given the expression \( \frac{x^{2}-2}{x^{2}+6x-7}+\frac{19-4x}{7-6x-x^{2}} \). We notice that both denominators can be factored. Factoring, we get \( \frac{x^{2}-2}{(x-1)(x+7)}+\frac{19-4x}{(1-x)(7-x)} \)

Step 2: Convert the negative denominator to its additive inverse

Notice that the denominators \( (x-1)(x+7) \) and \( (1-x)(7-x) \) are additive inverses. We can convert \( (1-x)(7-x) \) to \( -(x-1)(x+7) \) in the second fraction. This gives us the expression \( \frac{x^{2}-2}{(x-1)(x+7)}-\frac{19-4x}{(x-1)(x+7)} \)

Step 3: Add the fractions

Now that the denominators are the same, we can combine the numerators: \( \frac{x^{2}-2-(19-4x)}{(x-1)(x+7)} \)

Step 4: Simplify the fraction

Simplify the numerator: \( \frac{x^{2}-4x-21}{(x-1)(x+7)} \)

Key Concepts

Understanding Additive Inverses

Additive inverses are a fundamental concept in algebra, which particularly come into play when working with algebraic fractions. In essence, an additive inverse of a number is what you add to that number to get a sum of zero. For example, the additive inverse of 3 is -3 because when you add them together (3 + (-3)), the result is 0.

Let's look at how this applies to the denominators in our exercise. The denominators (x-1)(x+7) and (1-x)(7-x) when simplified, look very different, but they are actually additive inverses. The reason is: if we distribute the negative through (1-x)(7-x), we end up with (-1 + x)(-7 + x), which is just a negated form of (x-1)(x+7). Recognizing additive inverses allows us to combine fractions by either adding or subtracting them—as we do with these algebraic fractions—by converting one to match the other in sign and terms.

Factoring Polynomials

Factoring polynomials is an essential skill when simplifying algebraic fractions or solving equations. Factoring is the process of breaking down a complex expression into simpler parts, or 'factors', that when multiplied together will give you the original expression.

In the given exercise, both denominators are polynomials that can be factored. The quadratic polynomial (x^2+6x-7) can be factored into two binomials (x-1)(x+7). The process usually involves finding two numbers that both add up to the coefficient of the middle term (in this case, 6) and multiply to the constant term (in this case, -7). This successfully breaks down the complex denominator into a more manageable form that can be used to simplify the overall expression. Similarly, for the second denominator, we're looking for factors of the form (a-x)(b-x) which satisfy ab = -7 and a + b = -6—leading us to (1-x)(7-x) as the factorization of 7-6x-x^2.

Combining Like Terms

After finding a common denominator, combining like terms is the next step in simplifying algebraic fractions. Like terms are terms that have identical variable parts, such as 2x and −5x, or −3x^2 and 7x^2. They can be combined by adding or subtracting their coefficients.

For example, in our exercise, after ensuring the denominators are identical, we end up with (x^2 - 2) and (−19 + 4x) in the numerator when we combine the two fractions. To simplify this, we find and combine like terms, which means reordering and grouping the x-terms together and the constant terms together. We'll then add or subtract their coefficients to get a single expression: x^2 - 4x – 21. This is a much simpler form that clearly shows the relationship between the variable and the coefficients, and is easier to work with either for further simplification or for finding solutions to equations.