Questions: We want to solve the equation 2/(x-3)+3/(x-5)=(x+1)/((x-3)(x-5)). a. What is the common denominator we will create during the solution process? (x-3)(x-5) b. What values of x do we know CANNOT be solutions to this equation? Write your answers as a comma-separated list. x= c. The first few steps in the solution process are shown below. Fill in any missing parts in the steps. 2/(x-3) + 3/(x-5) = (x+1)/((x-3)(x-5)) 2/(x-3) * +3/(x-5) * =(x+1)/((x-3)(x-5)) 2(x-5)/((x-3)(x-5))+= (x+1)/((x-3)(x-5)) times=x+1 d. Complete the solution process and enter the solution(s) below. Include only actual solutions in your list - discard any extraneous solutions, If there are multiple solutions, enter them as a comma-separated list, If there are no solutions to the equation, choose the "no solution" option. x=

We want to solve the equation 2/(x-3)+3/(x-5)=(x+1)/((x-3)(x-5)).
a. What is the common denominator we will create during the solution process?
(x-3)(x-5)
b. What values of x do we know CANNOT be solutions to this equation? Write your answers as a comma-separated list.
x=
c. The first few steps in the solution process are shown below. Fill in any missing parts in the steps.
2/(x-3) + 3/(x-5) = (x+1)/((x-3)(x-5))
2/(x-3) * +3/(x-5) * =(x+1)/((x-3)(x-5))
2(x-5)/((x-3)(x-5))+= (x+1)/((x-3)(x-5))
times=x+1
d. Complete the solution process and enter the solution(s) below. Include only actual solutions in your list - discard any extraneous solutions, If there are multiple solutions, enter them as a comma-separated list, If there are no solutions to the equation, choose the "no solution" option.
x=

Transcript text: We want to solve the equation $\frac{2}{x-3}+\frac{3}{x-5}=\frac{x+1}{(x-3)(x-5)}$. a. What is the common denominator we will create during the solution process? \[ (x-3)(x-5) \] b. What values of $x$ do we know CANNOT be solutions to this equation? Write your answers as a comma-separated list. \[ x=\square \] c. The first few steps in the solution process are shown below. Fill in any missing parts in the steps. \[ \begin{array}{r} \frac{2}{x-3}+\frac{3}{x-5}=\frac{x+1}{(x-3)(x-5)} \\ \frac{2}{x-3} \cdot \square+\frac{3}{x-5} \cdot \square=\frac{x+1}{(x-3)(x-5)} \\ \frac{2(x-5)}{(x-3)(x-5)}+\square=\frac{x+1}{(x-3)(x-5)} \\ \times=x+1 \end{array} \] d. Complete the solution process and enter the solution(s) below. Include only actual solutions in your list - discard any extraneous solutions, If there are multiple solutions, enter them as a commaseparated list, If there are no solutions to the equation, choose the "no solution" option. $x=$ $\square$

Solution

Solution Steps

To solve the given equation, we need to follow these steps:

Identify the common denominator for the fractions.
Determine the values of $ x $ that make the denominators zero, as these cannot be solutions.
Multiply each term by the common denominator to eliminate the fractions.
Simplify and solve the resulting polynomial equation.
Check the solutions to ensure they do not make any denominators zero.

Step 1: Identify the Common Denominator

The common denominator for the fractions in the equation $\frac{2}{x-3} + \frac{3}{x-5} = \frac{x+1}{(x-3)(x-5)}$ is $(x-3)(x-5)$.

Step 2: Determine Values of $ x $ That Cannot Be Solutions

The values of $ x $ that make the denominators zero are $ x = 3 $ and $ x = 5 $. Therefore, these values cannot be solutions to the equation.

Step 3: Multiply Each Term by the Common Denominator

Multiplying each term by $(x-3)(x-5)$ to eliminate the fractions, we get: \[ 2(x-5) + 3(x-3) = x+1 \]

Step 4: Simplify and Solve the Resulting Polynomial Equation

Simplifying the equation: \[ 2x - 10 + 3x - 9 = x + 1 \] Combining like terms: \[ 5x - 19 = x + 1 \] Subtracting $ x $ from both sides: \[ 4x - 19 = 1 \] Adding 19 to both sides: \[ 4x = 20 \] Dividing by 4: \[ x = 5 \]

Step 5: Check the Solutions

We need to check if $ x = 5 $ is a valid solution. Since $ x = 5 $ makes the denominator zero, it is not a valid solution.