Problem 29 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{1}{x-1}+5=\frac{11}{x-1}$$

Step-by-Step Solution

Answer

The solution is None, as the equation does not have a valid solution.

Step 1: Simplify Both Sides of the Equation

Since \(\frac{1}{x-1}\) and \(\frac{11}{x-1}\) have the same denominator, we can combine these terms. This gives us: \(\frac{1+5}{x-1}=\frac{11}{x-1}\) which simplifies to \(\frac{6}{x-1}=\frac{11}{x-1}\).

Step 2: Cross Multiply

Clear the fractions by multiplying each side of the equation by its denominator. In this case, each term can be multiplied by x-1. This will yield \(6=x-1=11\).

Step 3: Simplify and Solve for x

The equation simplifies to \(6=11\), as we have cleared all variables. Since \(6 \neq 11\), the equation has no solution.

Step 4: Check the Solution

Even though we realize the equation has no solution at Step 3, it's crucial to make sure this does not cause a division by zero. Notice that \(x=1\) would make the denominator zero in the fractions in the original equation, hence it is an excluded value. This corroborates that the equation indeed has no solution. Therefore, the final solution is empty, or none.

Key Concepts

No Solution in Equations

In algebra, we sometimes encounter equations where it's impossible to find a solution. This might occur when the simplification leads to a logical contradiction like "6 = 11." Such contradictions often appear after steps of solving, and they signal that the equation has no solution.

This means there is no value for the variable that will satisfy the condition of the equation. It’s like trying to find a matching puzzle piece when there isn’t one available.

Determining an equation has no solution:

• Simplify the equation as much as possible.
• Look for any contradictions during the simplification process.
• Confirm that the equation can't be solved by checking for excluded values, which ensure there's no division by zero.

In the given problem, \(6 eq 11\) is a clear indication of no solution.

Cross Multiplication

Cross multiplication is a powerful method used in solving rational equations. It allows us to eliminate fractions and make equations easier to handle.

Here's how cross multiplication works:

• Identify the denominators of the rational expressions.
• Multiply the numerator of each side by the denominator of the other side.

In our example, both sides of the equation have the common denominator (x−1). So, we only need to multiply by \(x-1\) to clear the fraction, leading to an expression without denominators.

The result here was \(6 = 11\). Although this did not give a solution, cross multiplication helped us reach a point where we could clearly see that there's a contradiction.

Excluded Values

In the context of rational equations, excluded values are specific values for the variable that make any denominator in the equation equal to zero. Since division by zero is undefined, these values cannot be solutions for the equation.

Identifying excluded values:

• Look at each denominator in the original equation.
• Set each denominator equal to zero.
• Solve for the variable to find values that make the denominator zero.

In our example, the expression \(x-1\) appeared in the denominator. Setting \(x-1 = 0\) gives \(x=1\), indicating an excluded value.

By being aware of excluded values, we ensure that our solutions do not lead to division by zero, maintaining mathematical correctness.