$\frac{1}{x+1}+4=\frac{9}{x+1}$
a. What is/are the value or values of the variable that make(s) the denominators zero?
$x=\square$
(Simplify your answer. Use a comma to separate answers as needed.)
b. Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is $\square$}.
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
B. The solution set is ${x \mid x$ is a real number $}$.
C. The solution set is $\varnothing$.
Answer (2)
Find the value of x that makes the denominator zero: x = − 1 .
Multiply both sides of the equation by ( x + 1 ) to eliminate the denominators.
Simplify the equation and solve for x : x = 1 .
The solution set is 1 .
Explanation
Problem Analysis We are given the equation x + 1 1 + 4 = x + 1 9 . We need to find the value(s) of x that make the denominator zero and then solve the equation for x .
Finding Values That Make Denominator Zero a. To find the value(s) of x that make the denominator zero, we set the denominator equal to zero and solve for x :
x + 1 = 0 x = − 1 So, the value of x that makes the denominator zero is x = − 1 .
Solving the Equation b. To solve the equation, we first multiply both sides of the equation by ( x + 1 ) to eliminate the denominators: ( x + 1 ) ( x + 1 1 + 4 ) = ( x + 1 ) ( x + 1 9 ) 1 + 4 ( x + 1 ) = 9 Now, we simplify and solve for x :
1 + 4 x + 4 = 9 4 x + 5 = 9 4 x = 9 − 5 4 x = 4 x = 4 4 x = 1 Now, we check if this solution is valid. Since x = 1 does not make the denominator zero, it is a valid solution.
Final Answer Therefore, the solution set is 1 .
Examples
Understanding how to solve rational equations is crucial in many real-world applications, such as calculating the flow rate in pipes or determining the concentration of a substance in a mixture. For instance, if you're designing a pipeline system, you might use rational equations to model the flow of fluids through different sections of the pipe. By solving these equations, you can optimize the pipe diameters and flow rates to ensure efficient and safe operation of the system. Similarly, in chemistry, you might use rational equations to calculate the equilibrium concentrations of reactants and products in a chemical reaction. These calculations help chemists understand and control chemical processes, leading to the development of new materials and technologies.
The value of x that makes the denominator zero is − 1 . The solution to the equation is x = 1 , therefore the solution set is {1}.
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