Problem 84 - College Algebra

Question

Which one of the following is true? a. \(\frac{x^{2}-25}{x-5}=x-5\) b. \(\frac{x}{y} \div \frac{y}{x}=1,\) if \(x \neq 0\) and \(y \neq 0\) c. The least common denominator needed to find \(\frac{1}{x}+\frac{1}{x+3}\) is \(x+3\) d. The rational expression $$\frac{x^{2}-16}{x-4}$$ is not defined for \(x=4 .\) However, as \(x\) gets closer and closer to \(4,\) the value of the expression approaches \(8 .\)

Step-by-Step Solution

Answer

The true statement among the given options is D.

Step 1: Statement A Verification

Firstly, factorize the numerator in \(\frac{x^{2}-25}{x-5}\) as \((x-5)(x+5)\). Hence, \(\frac{x^{2}-25}{x-5}=x+5\). So, statement A, which states \(\frac{x^{2}-25}{x-5}=x-5\), is not correct.

Step 2: Statement B Verification

Using division rules for fractions, we rewrite \(\frac{x}{y} \div \frac{y}{x}\) as \(\frac{x}{y} \times \frac{x}{y}\) and the result is \(\frac{x^2}{y^2} \neq 1\). Hence, statement B is also incorrect assuming \(x \neq y\). If \(x = y\), this would be true. The question doesn't provide this information, so we cannot surely say this statement is true.

Step 3: Statement C Verification

The least common denominator (LCD) between \(x\) and \(x+3\) is \(x(x+3)\), so the statement C, which states LCD is \(x+3\), is incorrect.

Step 4: Statement D Verification

Firstly, factorize the numerator in \(\frac{x^{2}-16}{x-4}\) as \((x-4)(x+4)\). Then, \(\frac{x^{2}-16}{x-4} = x+4\). The statement D claims that this expression approaches 8 as \(x\) gets closer to 4, but we see it actually approaches \(4+4 = 8\). So statement D is correct.

Key Concepts

Factoring

Factoring is a crucial process in simplifying rational expressions. Consider breaking down the numerator or the denominator into simpler expressions that multiply to give the original term. Let's look at an example from the exercise.In the expression \(\frac{x^{2} - 25}{x - 5}\), the numerator \(x^2 - 25\) is a difference of squares. When you factor it, you get \((x - 5)(x + 5)\). This helps us reduce or simplify the rational expression by canceling like terms if possible.Here are some steps to follow when factoring:

• Identify common factors: Look for terms that can be factored out from the expression.
• Recognize patterns such as difference of squares, perfect square trinomials, or cubes.
• Rewrite the expression: Once you factor, rewrite the expression to see if any terms can be canceled.

This process is essential in simplifying rational expressions and is often used in conjunction with other algebraic rules.

Least Common Denominator

When working with rational expressions, finding the Least Common Denominator (LCD) is necessary to combine fractions. The LCD is the smallest expression that both denominators divide without leaving a remainder.In the exercise, we focus on combining fractions like \(\frac{1}{x} + \frac{1}{x+3}\). The proposed LCD is \(x + 3\), which is incorrect. The correct LCD should be \(x(x + 3)\). Here's why:

• The LCD must be a multiple of all denominators involved. For \(\frac{1}{x}\) and \(\frac{1}{x+3}\), no single expression \(x\) or \(x+3\) achieves this.
• Multiply the denominators: By multiplying each distinct denominator term, \(x(x + 3)\), you form the smallest possible expression that all denominators divide.
• Adjust numerators accordingly: Express each fraction using the LCD, allowing you to add or subtract them easily.

Understanding LCD is key in solving problems involving addition or subtraction of fractions in algebra.

Division of Fractions

The division of fractions can often confuse students, yet it relies on a simple rule: when you divide by a fraction, you multiply by its reciprocal.In the exercise, it was important to verify statement \(b\), which states \(\frac{x}{y} \div \frac{y}{x}\). To solve this division, rewrite it as \(\frac{x}{y} \times \frac{x}{y}\). This changes the operation from division to multiplication by flipping the second fraction.Here's a quick guide to dividing fractions:

• Flip the divisor (the fraction you are dividing by) to find its reciprocal.
• Multiply the numerators together and the denominators together.
• Simplify the resulting fraction if possible.

In the exercise, the expression yields \(\frac{x^2}{y^2}\), which shows statement \(b\) doesn't hold true unless \(x = y\). Understanding this process is critical for mastering the division of rational expressions.