Problem 18 - College Algebra

Question

For the following exercises, solve each rational equation for \(x\). State all \(x\) -values that are excluded from the solution set. $$ \frac{3}{x-2}=\frac{1}{x-1}+\frac{7}{(x-1)(x-2)} $$

Step-by-Step Solution

Answer

The solution is \(x = 4\), with exclusions \(x \neq 1, 2\).

Step 1: Determine the excluded values

The expressions in the denominators are \(x-2\) and \( (x-1)(x-2) \). To determine the excluded values, set \(x-2=0\) and \(x-1=0\) separately to avoid division by zero. Solving these gives \(x=2\) and \(x=1\). Therefore, \(x\) cannot be 1 or 2.

Step 2: Clear the denominators

Multiply every term of the equation by the least common denominator (LCD), which is \((x-1)(x-2)\) to eliminate fractions:\[(x-1)(x-2) \cdot \frac{3}{x-2} = (x-1)(x-2) \cdot \frac{1}{x-1} + (x-1)(x-2) \cdot \frac{7}{(x-1)(x-2)}\]This simplifies to:\[3(x-1) = (x-2) + 7\]

Step 3: Simplify and solve the equation

Distribute and simplify the equation:\[3x - 3 = x - 2 + 7\]Combine like terms on the right side:\[3x - 3 = x + 5\]Move \(x\) terms to one side and constants to the other:\[3x - x = 5 + 3\]Simplify:\[2x = 8\]Divide both sides by 2:\[x = 4\]

Step 4: Verify the solution against the excluded values

Check that the solution \(x = 4\) is not one of the excluded values \(x = 1\) or \(x = 2\). Since it is not, \(x = 4\) is a valid solution.

Key Concepts

Excluded Values

In rational equations, the denominators consist of polynomial expressions. However, when these denominators are zero, the expression becomes undefined due to division by zero. That's where the concept of excluded values comes into play. Excluded values are those specific values of the variable that make any denominator in the rational equation equal to zero, thus making the equation undefined.

To identify excluded values, you set each denominator equal to zero and solve for the variable. In our example equation, we have the denominators \(x - 2\) and \((x - 1)(x - 2)\). To find the excluded values, we solve the following equations:

• \(x - 2 = 0\) leading to \(x = 2\)
• \(x - 1 = 0\) leading to \(x = 1\)

These values of \(x\) are excluded from the solution set because they make the denominator zero, causing an undefined expression.

Least Common Denominator

The least common denominator (LCD) is a crucial tool in solving rational equations. It serves as a universal denominator for the equation, allowing the elimination of fractions by transforming them into whole numbers. This is essential in rational equations where multiple fractions are present with different denominators.

To find the LCD, take the product of all distinct denominators in the equation, making sure each distinct factor appears with its highest power that appears among the denominators. In our example, the denominators are \(x - 2\) and \((x - 1)(x - 2)\). Hence, the LCD is \((x - 1)(x - 2)\) since it includes all factors of the individual denominators. By multiplying every term of the equation by the LCD, all fractions are cleared, greatly simplifying the process of solving the rational equation.

Clearing Fractions

Clearing fractions is a technique used to simplify the process of solving rational equations. Once you have determined the least common denominator, you can use it to "clear" the fractions from the equation by multiplying every term by the LCD. This converts the equation into a more straightforward form without fractions.

In the given equation, every term is multiplied by \((x - 1)(x - 2)\)—the LCD—to eliminate the fractions:

• For \(\frac{3}{x-2}\), multiply to get \(3(x - 1)\)
• For \(\frac{1}{x-1}\), multiply to get \(x - 2\)
• For \(\frac{7}{(x-1)(x-2)}\), it simplifies directly to \(7\) since it's already the LCD

The result is a simple linear equation \(3(x - 1) = (x - 2) + 7\), which is much easier to solve. Simplifying and solving this equation leads to finding possible solutions while always keeping in mind the excluded values.