Problem 112 - College Algebra with Modeling and Visualization

Question

Clear fractions and solve. $$ \frac{1}{x-2}+\frac{1}{x-3}-\frac{2}{x}=0 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{12}{5}\).

Step 1: Identify the Least Common Denominator (LCD)

To clear the fractions, we first need to identify the least common denominator (LCD) of the given fractions. The denominators are \(x-2\), \(x-3\), and \(x\). So, the LCD is \((x-2)(x-3)x\).

Step 2: Multiply Each Term by the LCD

Multiply each term of the equation by \((x-2)(x-3)x\) to eliminate the fractions:\[(x-2)(x-3)x \cdot \frac{1}{x-2} + (x-2)(x-3)x \cdot \frac{1}{x-3} - (x-2)(x-3)x \cdot \frac{2}{x} = 0\]

Step 3: Simplify Each Term

After multiplying, simplify each term:- The first term simplifies to \((x-3)x\).- The second term simplifies to \((x-2)x\).- The third term simplifies to \(2(x-2)(x-3)\).This gives us the equation:\[(x-3)x + (x-2)x - 2(x-2)(x-3) = 0\]

Step 4: Expand and Combine Like Terms

Expand the terms and combine like terms:\[x^2 - 3x + x^2 - 2x - 2(x^2 - 5x + 6) = 0\]Combine:\[2x^2 - 5x - 2x^2 + 10x - 12 = 0\]

Step 5: Simplify to a Solvable Equation

Further simplify the equation:\[5x - 12 = 0\]

Step 6: Solve for x

Isolate the variable \(x\) by adding 12 to both sides:\[5x = 12\]Then, divide by 5:\[x = \frac{12}{5}\]

Key Concepts

Least Common Denominator

When working with equations containing fractions, it's crucial to identify a common ground so that the fractions can be easily manipulated or eliminated. This common ground is the "Least Common Denominator" (LCD). The LCD is the smallest expression that all the denominators of an equation share as a factor.

In our example, we are dealing with three fractions with denominators: \(x-2\), \(x-3\), and \(x\). To find the LCD, we simply multiply these expressions together to form \((x-2)(x-3)x\).

• This step is important because it allows us to rewrite the equation without fractions.
• By multiplying every term in the equation by the LCD, we ensure each fraction is "cleared" because the denominators divide perfectly into it.

A common mistake is forgetting to multiply every single term in the equation, not just the fractions themselves. The LCD helps in converting the equation into a simpler form, which is much easier to solve.

Solving Equations

Solving equations is all about finding the value of the variable that makes the equation true. After eliminating the fractions using the LCD, you're left with a polynomial equation. This is a key step as it transforms the equation into something more manageable.

Let's look at the process:

• Begin by simplifying the equation if possible just as in our example. Here, multiplying each term by the LCD allows us to cancel out denominators.
• Once the denominators are gone, simplify by expanding terms and combining like terms, as shown when we get the equation \[(x-3)x + (x-2)x - 2(x-2)(x-3) = 0\].
• Next, streamline this equation further until the equation isolates the variable, as we ended with \[5x - 12 = 0\].

This problem-solving method is effective for rational equations, and the steps ensure clarity and reduce possible errors. Always re-check each step for accuracy.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both, contain algebraic expressions. They can be tricky, but understanding their behavior follows the same principles as simple numeric fractions.

Key points to remember include:

• The operations for simplifying or finding the LCD don't change; they rely on factoring and simplifying algebraic expressions just like numeric fractions do.
• When solving equations, it's crucial to make sure the denominator never equals zero, as fractions with zero in the denominator are undefined.
• After clearing the fractions by finding the LCD, treat the remaining equation as you would a standard polynomial equation. This might involve expanding brackets or using basic algebraic operations to collect like terms.

In our specific exercise, understanding algebraic fractions is fundamental to properly apply the LCD and make computations that lead to solving the original equation. They may appear complicated, but by approaching them systematically, you can manage them just like any other algebraic problem.