Which statement describes the solutions of this equation?
[tex]\frac{x}{3}+\frac{x-1}{x+4}=\frac{x+3}{x+4}[/tex]
A. The equation has no valid solutions and two extraneous solutions.
B. The equation has one valid solution and no extraneous solutions.
C. The equation has one valid solution and one extraneous solution.
D. The equation has two valid solutions and no extraneous solutions.
Answer (1)
Identify restrictions: x = − 4 .
Simplify the equation to 3 x = x + 4 4 .
Cross-multiply and rearrange to get x 2 + 4 x − 12 = 0 .
Factor the quadratic: ( x + 6 ) ( x − 2 ) = 0 , yielding solutions x = − 6 and x = 2 . Both are valid.
The final answer is D: The equation has two valid solutions and no extraneous solutions.
Explanation
Identify Restrictions We are given the equation 3 x + x + 4 x − 1 = x + 4 x + 3 and asked to determine the nature of its solutions (valid or extraneous). First, we need to identify any restrictions on x due to the denominators. Since we have x + 4 in the denominator, x = − 4 .
Isolate x Next, we want to isolate x . Subtract x + 4 x − 1 from both sides of the equation: 3 x = x + 4 x + 3 − x + 4 x − 1 Since the denominators on the right side are the same, we can combine the fractions: 3 x = x + 4 ( x + 3 ) − ( x − 1 ) Simplify the numerator: 3 x = x + 4 x + 3 − x + 1 = x + 4 4
Cross-Multiply Now, we cross-multiply to eliminate the fractions: x ( x + 4 ) = 3 ( 4 ) x 2 + 4 x = 12
Quadratic Form Rearrange the equation into a standard quadratic form: x 2 + 4 x − 12 = 0
Factor the Quadratic We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to -12 and add to 4. These numbers are 6 and -2. Thus, we can factor the quadratic as: ( x + 6 ) ( x − 2 ) = 0
Solve for x This gives us two possible solutions for x :
x + 6 = 0 ⇒ x = − 6 x − 2 = 0 ⇒ x = 2
Check for Extraneous Solutions Now we need to check if these solutions are valid or extraneous. Recall that we have the restriction x = − 4 . Since neither of our solutions is -4, both are valid.
Therefore, the equation has two valid solutions: x = − 6 and x = 2 .
Final Answer The equation has two valid solutions, x = − 6 and x = 2 , and no extraneous solutions. Therefore, the correct statement is D.
Examples
When designing a bridge, engineers often use rational equations to model the forces and stresses acting on different parts of the structure. Solving these equations helps them determine the optimal dimensions and materials to ensure the bridge's stability and safety. For example, the equation 3 x + x + 4 x − 1 = x + 4 x + 3 could represent a simplified model of how loads are distributed across different sections of the bridge. Finding the valid solutions for x would then provide critical information about the load-bearing capacity at specific points.
