Problem 6 - Intermediate Algebra

Question

To clear the following equations of fractions, by what should both sides be multiplied? a. \(\frac{1}{a}=\frac{1}{3}-\frac{2}{3 a}\) b. \(\frac{2}{x-2}+\frac{10}{x+5}=\frac{2 x}{x^{2}+3 x-10}\)

Step-by-Step Solution

Answer

a. Multiply each side by \(3a\). b. Multiply each side by \((x-2)(x+5)\).

Step 1: Identify the Denominators

For part a, the fractions are \(\frac{1}{a}\), \(\frac{1}{3}\), and \(\frac{2}{3a}\). Thus, the denominators are \(a\), 3, and \(3a\). In part b, the fractions are \(\frac{2}{x-2}\), \(\frac{10}{x+5}\), and \(\frac{2x}{x^2+3x-10}\). Therefore, the denominators are \(x-2\), \(x+5\), and the expression \(x^2+3x-10\).

Step 2: Find the Least Common Denominator (LCD) - Part a

For part a, the LCD should include all prime factors from each denominator: \(a\), 3, and \(3a\). The simplest LCD that includes these is \(3a\).

Step 3: Find the Least Common Denominator (LCD) - Part b

For part b, we need to factor the quadratic expression in the denominator. The expression \(x^2+3x-10\) factors to \((x-2)(x+5)\). Therefore, the complete LCD is obtained by combining all distinct factors: \((x-2)(x+5)\).

Step 4: Determine the Multiplying Factor - Part a

In order to eliminate the fractions in part a, multiply all terms of the equation by the LCD, \(3a\). This will clear the fractions.

Step 5: Determine the Multiplying Factor - Part b

To clear the fractions in part b, multiply all terms of the equation by the expression \((x-2)(x+5)\). This ensures that every fraction will have its denominator canceled with this factor.

Key Concepts

Understanding the Least Common Denominator

In solving rational equations, identifying the Least Common Denominator (LCD) is a critical step. The LCD is the smallest expression that all denominators can divide into evenly. To find the LCD:

• List all denominators in your equation. For example, in equation (a), we have the denominators \(a\), 3, and \(3a\).
• Identify the distinct factors among these denominators. Here they are \(a\) and 3.
• The LCD should include each distinct factor at its highest power appearing in any of the denominators.

For part (a), we choose \(3a\) as the LCD since it covers each denominator factor. In part (b), the expression involves factoring the quadratic \(x^2 + 3x - 10\) into \((x-2)(x+5)\). Therefore, the LCD combines these with other terms such as \(x-2\) and \(x+5\). Hence, \((x-2)(x+5)\) becomes the LCD for equation (b). Ensuring that each term is included at its necessary power is key to finding the correct LCD.

Clearing Fractions in Equations

Once you find the LCD, the next step is clearing the fractions. Clearing fractions involves multiplying every term of the equation by the LCD. This step eliminates all fractions, making the equation simpler and more manageable.Here's how it works:

• Multiply each term in the equation by the identified LCD.
• This action cancels out the denominators across the terms.

In practice, for part (a), since our LCD is \(3a\), multiply every term including the constant on both sides by \(3a\). For part (b), multiply all terms by \((x-2)(x+5)\), the LCD. By doing this, each denominator will divide evenly, rendering it reduced to 1.This approach ensures that you end up with a fraction-free equation, paving the way for simpler arithmetic and solutions.

Factoring Quadratics

Factoring quadratics is a crucial skill when dealing with more complex denominators in rational equations. Quadratics appear frequently but need to be simplified by factoring them where possible. Consider part (b) of the original exercise.The denominator \(x^2 + 3x - 10\) is a quadratic expression. To factor it, identify two numbers that both:

• Multiply to the constant term (-10).
• Add to the linear coefficient (+3).

For \(x^2 + 3x - 10\), these numbers are 5 and -2. Therefore, it factors into \((x-2)(x+5)\). This factorization helps construct the LCD, which ensures efficient clearing of fractions.Once factored, this expression assists in determining the overall LCD and streamlining the process of solving for variables within rational equations. Understanding how to factor accurately is essential for maintaining clarity and accuracy in your mathematical solutions.