Problem 19 - College Algebra

Question

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

Step-by-Step Solution

Answer

No valid solutions since \(x = 1\) is excluded.

Step 1: Identify the common denominator

The denominators in the equation are \(x-1\). Find the Least Common Denominator (LCD), which is \(x-1\).

Step 2: Eliminate the denominators

Multiply every term in the equation by the LCD \((x-1)\) to eliminate the fractions:\[(x-1) \left( \frac{3x}{x-1} \right) + (x-1)(2) = (x-1) \left( \frac{3}{x-1} \right)\]This simplifies to:\[3x + 2(x-1) = 3\]

Step 3: Simplify the equation

Distribute the 2 in the term \(2(x-1)\):\[3x + 2x - 2 = 3\]Combine like terms to get:\[5x - 2 = 3\]

Step 4: Solve for \(x\)

Add 2 to both sides of the equation:\[5x = 5\]Then divide by 5:\[x = 1\]

Step 5: State excluded values

The original equation has a denominator of \(x-1\). Since a denominator cannot be 0, \(x = 1\) is excluded from the solution set.

Key Concepts

Solving Rational Equations

Rational equations are equations that include fractions with polynomials in the numerator or the denominator. The journey to solving rational equations starts with identifying and working with these fractions. To solve a rational equation such as \( \frac{3x}{x-1} + 2 = \frac{3}{x-1} \), the core approach is to eliminate the pesky denominators that make things complex. This is achieved by multiplying every term by the least common denominator (LCD) effectively making the equation simpler by removing fractions.

Once the denominators are out of the way, we handle the resulting equation like any other algebraic one. Simplify it, combine like terms, and solve for the variable. This process transforms the equation so it's easier to work with and leads directly to finding the solution for \( x \). Remember, solving such equations is just unwrapping a complicated present until we reach the core solution.

Excluded Values in Equations

While solving rational equations, it is essential to identify the values that are not allowed in the solution set. These are often called excluded values. They appear because certain values make the denominator zero, which is not permitted since division by zero is undefined.

In our example, the equation \( \frac{3x}{x-1} + 2 = \frac{3}{x-1} \) has a denominator \( x-1 \), which becomes zero when \( x = 1 \). Therefore, even though mathematically you might find \( x = 1 \) while solving, it needs to be excluded from the solution set. This means any solution that makes the denominator of the original equation zero must be left out, ensuring our answers are valid without leading to undefined expressions.

Least Common Denominator (LCD)

Finding and using the least common denominator (LCD) is a crucial step in solving rational equations. The LCD is the smallest expression that can act as a denominator for all the fractions involved in the equation. It helps to clear out the fractions, simplifying the process significantly.

In our case with \( \frac{3x}{x-1} + 2 = \frac{3}{x-1} \), the LCD is \( x-1 \) because it covers both fractional terms. By multiplying each term of the equation by \( x-1 \), we effectively remove the denominators, translating the equation into a much simpler linear form. Utilizing the LCD not only avoids fractional headaches but also streamlines solving the equation, making math feel less of a hurdle.