Problem 71 - Algebra for College Students

Question

For which real numbers does \(\frac{x}{x-3}+\frac{4}{x}\) equal \(\frac{(x+6)(x-2)}{x(x-3)} ?\) Explain your answer.

Step-by-Step Solution

Answer

Real numbers excluding \( x = 0 \) and \( x = 3 \).

Step 1: Identify common denominators

To solve the equation \( \frac{x}{x-3} + \frac{4}{x} = \frac{(x+6)(x-2)}{x(x-3)} \), we first identify the least common denominator (LCD) for the left-hand side fractions. The LCD is \( x(x-3) \), which is already the denominator on the right-hand side.

Step 2: Combine left-hand side fractions

Combine the fractions on the left-hand side to form a single fraction over the common denominator: \( \frac{x}{x-3} + \frac{4}{x} \) becomes \( \frac{x \cdot x + 4(x-3)}{x(x-3)} = \frac{x^2 + 4x - 12}{x(x-3)} \).

Step 3: Set numerators equal

With the common denominator in place, set the numerators of the equation equal: \( x^2 + 4x - 12 = (x+6)(x-2) \). Expand the right-hand side to get \( x^2 + 4x - 12 = x^2 + 4x - 12 \). This equality is true for all \( x \) except where the denominators are zero.

Step 4: Determine restrictions on x

Examine the denominators: \( x(x-3) \). Set each factor to zero: \( x = 0 \) or \( x - 3 = 0 \), which shows \( x = 3 \). Since division by zero is undefined, \( x eq 0 \) and \( x eq 3 \).

Step 5: Write the solution

The solution to the equation is all real numbers except \( x = 0 \) and \( x = 3 \), where division by zero would occur.

Key Concepts

Least Common Denominator (LCD)

The concept of the Least Common Denominator (LCD) plays a crucial role when working with rational equations. The LCD is the smallest expression that each of the denominators in a set can divide without leaving a remainder. For the equation \( \frac{x}{x-3} + \frac{4}{x} = \frac{(x+6)(x-2)}{x(x-3)} \), the denominators on the left-hand side are \( x-3 \) and \( x \). To add these fractions, you need the LCD, which is \( x(x-3) \). This denominator ensures that both fractions have the same base, which is necessary for addition. The right-hand side already has this as its denominator, making it convenient to combine the fractions on the left.

Combining Fractions

Once you have determined the Least Common Denominator (LCD), you can combine fractions. Combining fractions involves rewriting them with the same denominator, so that terms can be added or subtracted easily.For \( \frac{x}{x-3} + \frac{4}{x} \), convert each fraction to have the common denominator \( x(x-3) \). This results in:

• \( \frac{x}{x-3} = \frac{x \cdot x}{x(x-3)} = \frac{x^2}{x(x-3)} \)
• \( \frac{4}{x} = \frac{4(x-3)}{x(x-3)} = \frac{4x - 12}{x(x-3)} \)

Add these new fractions:\[ \frac{x^2}{x(x-3)} + \frac{4x - 12}{x(x-3)} = \frac{x^2 + 4x - 12}{x(x-3)} \]With the fractions combined, we now have a single rational expression on the left-hand side to match the structure on the right.

Restrictions on Variables

When solving rational equations, it's vital to determine restrictions on the variables present. Restrictions often arise from the need to avoid division by zero, as rational expressions become undefined when their denominators are zero.In the equation \( \frac{x^2 + 4x - 12}{x(x-3)} = \frac{(x+6)(x-2)}{x(x-3)} \), the denominator is \( x(x-3) \). To find possible restrictions:

• Set each factor of the denominator equal to zero: \( x = 0 \) and \( x - 3 = 0 \).
• Solve for \( x \): \( x = 0 \) and \( x = 3 \).

These solutions correspond to restrictions on \( x \), making the original equation undefined at these points. Therefore, valid solutions can't include \( x = 0 \) or \( x = 3 \), since these would result in division by zero.

Division by Zero

Division by zero is a fundamental concept that must be avoided in mathematics. Division by zero occurs when an expression or equation attempts to divide a number by zero, which is undefined in the set of real numbers.In rational equations, division by zero poses a problem because it makes entire expressions meaningless or undefined. This is why we carefully examine the denominators of rational expressions for potential zero values. For instance, in the prior equation, we identified that \( x = 0 \) and \( x = 3 \) make the denominators zero. This principle guides us:

• We can't allow values that yield zero in any denominator.
• These restrictions must be listed alongside any solution to indicate where the equation does not hold.

Avoiding division by zero ensures that solutions to equations are valid and applicable.