$\frac{1}{x+1}+4=\frac{9}{x+1}$
a. What is/are the value or values of the variable that make(s) the denominators zero?
$x=1$
(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 {1}.
$\square$ .
(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 x + 1 equal to zero: x = − 1 .
Solve the equation x + 1 1 + 4 = x + 1 9 by isolating x .
Combine like terms: 4 = x + 1 8 .
Solve for x : x = 1 . The solution set is { 1 } .
The final answer is { 1 } .
Explanation
Understanding the Problem We are given the equation x + 1 1 + 4 = x + 1 9 . Part (a) asks for the value(s) of x that make the denominator zero. Part (b) asks us to solve the equation for x .
Finding Values That Make the Denominator Zero For part (a), the denominator is x + 1 . We need to find the value of x that makes x + 1 = 0 . Subtracting 1 from both sides, we get x = − 1 . So, x = − 1 makes the denominator zero.
Solving the Equation For part (b), we solve the equation x + 1 1 + 4 = x + 1 9 . First, subtract x + 1 1 from both sides of the equation: 4 = x + 1 9 − x + 1 1 Combine the fractions on the right side: 4 = x + 1 9 − 1 = x + 1 8 Multiply both sides by x + 1 : 4 ( x + 1 ) = 8 Divide both sides by 4: x + 1 = 4 8 = 2 Subtract 1 from both sides: x = 2 − 1 = 1
Checking the Solution Now we need to check if the solution x = 1 is valid. The denominator is x + 1 . If x = 1 , then the denominator is 1 + 1 = 2 , which is not zero. So, x = 1 is a valid solution. However, we must also consider the value x = − 1 , which makes the denominator zero. Since division by zero is undefined, x cannot be − 1 . Therefore, x = − 1 is not a solution to the equation.
Final Answer The solution to the equation is x = 1 . Therefore, the solution set is { 1 } .
Examples
When solving equations involving rational expressions, it's crucial to identify values that make the denominator zero, as these values are excluded from the solution set. This concept is applicable in various fields, such as physics when dealing with formulas involving division, or in engineering when designing systems where certain parameters cannot be zero to avoid system failure. Understanding these restrictions ensures accurate and meaningful results in practical applications.
The value that makes the denominator zero is x = − 1 . The solution to the equation is x = 1 , making the solution set {1}. Therefore, the correct choice is A: The solution set is {1}.
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